# Thermodynamics and Dynamic Stability: Extended Theories of Heat Conduction

R. Somogyfoki  Department of Theoretical Physics, Institute of Particle and
Nuclear Physics, HUN-REN Wigner Research Centre for Physics, Konkoly Thege
Miklós út. 29-33, Budapest H-1525, Hungary  ELTE Doctoral School of Physics,
Eötvös Loránd University, Pázmány Péter stny. 1/A, Budapest 1117, Hungary
Montavid Thermodynamic Research Group, Budapest, Hungary  A. Famà  Department
of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences,
University of Messina, Messina, Italy  L. Restuccia  Department of
Mathematical and Computer Sciences, Physical Sciences and Earth Sciences,
University of Messina, Messina, Italy  P. Ván  Department of Theoretical
Physics, Institute of Particle and Nuclear Physics, HUN-REN Wigner Research
Centre for Physics, Konkoly Thege Miklós út. 29-33, Budapest H-1525, Hungary
Department of Energy Engineering, Faculty of Mechanical Engineering, Budapest
University of Technology and Economics, Műgyetem rkp. 3, Budapest 1111,
Hungary  Montavid Thermodynamic Research Group, Budapest, Hungary

###### Abstract

The fundamental dynamic stability of heat conduction theories beyond Fourier
is analyzed in the framework of nonequilibrium thermodynamics. It is shown,
that the thermodynamic framework, concave entropy and nonnegative entropy
production, can ensure the stability of homogeneous thermodynamic equilibrium.
Various special heat conduction theories of Extended Thermodynamics are
compared and analysed in the general framework.

##  1 Introduction

Thermodynamics and stability are closely connected. The tendency to
homogeneous equilibrium in a neutral environment is our most general
expectation regarding macroscopic stability of materials. According to the
history of thermodynamics, this general experience was distilled in the Second
Law [1]. However, classical thermodynamics is a theory of homogeneous bodies,
and, in spite of its name, is not a dynamical theory. Thermodynamic processes
are different from mechanical ones: the classical theory of heat engines is
not like the Newtonian theory of planet motion. Therefore, without evolution
equations, the dynamic content of stability is a hidden aspect of the
classical theory. The origin of this problematic situation is partially
historical: the mathematical concepts of stability were missing when the
thermodynamic ones were already clarified. The concept of Lyapunov stability
appeared only in 1892 [2]. At that time the second and first laws were
established knowledge in a formalism of equilibrium thermodynamics with a
process concept but without time.

The first spacetime formulation appeared only in 1940, in the framework of
classical field theories when the Second Law of Thermodynamics was connected
to the fundamental continuum balances with the calculation of entropy
production. Classical Irreversible Thermodynamics, the classical field theory,
was born at the same time for nonrelativistic and relativistic fluids [3, 4].
Then the main research goal was the extension of the validity range
considering memory and inertial effects, leading to theories of Extended
Thermodynamics [5, 6, 7]. These theories introduced the dissipative
thermodynamics fluxes of Classical Irreversible Thermodynamics as
thermodynamics state variables [8]. The primary benchmark was the
compatibility with microscopic theories, first of all with kinetic theory of
rarefied gases [9, 10]. The question of stability appeared in two contexts.
First, the concept of dissipative structures based on the Second Law and on
the entropy balance [11]; and also the mathematical conditions of extending
the concepts of Lyapunov stability to inhomogeneous equilibrium solutions of
nonlinear partial differential equations [12, 13]. However, it was clear from
the beginning that dissipative structures appear in open systems, therefore
the fundamental stability of thermodynamic equilibrium is not an issue in this
respect, it was and still is accepted as an evident intuitive consequence of
the Second Law. On the other hand, mathematical investigations are struggling
with the proper interpretation of the physical concepts, in particular the
Second Law [14, 15]. For example, considering the entropy production as a
candidate of Lyapunov functional of homogeneous stationary states [11], is a
misleading endeavour.

The analogy between the properties of entropy and the properties of a Lyapunov
function is clear: entropy is concave and increasing along processes like a
Lyapunov function. However, there is a twofold conceptual difficulty: for
continuum theories the mathematics of Lyapunov stability is complicated, on
the other hand, in thermodynamics of homogeneous bodies there is no time,
there is no dynamics. Moreover, the attempts to understand the transition from
continuum to homogeneous through various homogenisation methods encounter ill-
defined process and equilibrium definitions [16, 17, 18].

The formulation of thermodynamics of homogeneous bodies – traditionally and
misleadingly called equilibrium thermodynamics – in a dynamical system
framework was possible only with a proper definition of reservoir entropies
and with the understanding that the simplest dynamics requires open systems, a
body-reservoir connection. The modeling framework may be called ordinary
thermodynamics because it reformulates the complete theory of Gibbsian
equilibrium thermodynamics in a dynamical system framework and also because it
is based on well-defined dynamical laws of ordinary differential equations
[19].

Ordinary thermodynamics is a theory of stability: the seemingly independent
physical postulates are unified in a mathematically consistent stability
framework with a clear physical meaning [20, 21, 22]. Then a question emerges
whether and how the continuum theory of nonequilibrium thermodynamics – with
partial differential equations – is connected to the ordinary differential
equations of ordinary thermodynamics? Whether the thermodynamic compatibility
of the constitutive equations of a continuum theory would ensure the stability
of homogeneous thermodynamic equilibrium or not. This question is particularly
interesting in the light of the recent development of nonequilibrium
thermodynamics, where the field equations are constructed with thermodynamic
principles, like in Extended Thermodynamics and in case of Internal Variables
[10, 9, 23, 24]. It is particularly important to analyse the relation of
universal theories (like gravity or quantum physics) and thermodynamic
principles. The stability background of thermodynamics could be an explanation
of their apparent universality.

A simple necessary stability condition is provided by the dispersion
relations: the conditions of the linear stability of homogeneous equilibrium
can be calculated. It is far from being a pre-played game, there are several
examples when stability of homogeneous equilibrium is expected, but not
fulfilled. The most notable example is the theory of one component dissipative
relativistic fluids, which is a major challenge of thermodynamics as a
stability theory. It is proved that the homogeneous equilibrium of first-order
dissipative relativistic fluids is unstable [25]. The instability disappears
in Landau–Lifshitz frame, and according to detailed analysis it is clearly
connected to the energy transport, to heat conduction [26]. The connection of
thermodynamics and stability of homogeneous equilibrium for first-order
dissipative relativistic fluids is an open problem. Recent investigations
suggest a special choice of flow-frames that ensure the expected stability of
homogeneous equilibrium, at least in some special cases, [27, 28, 29]. The
relativistic motivation of Extended Thermodynamics promised to restore the
fundamental stability properties of relativistic fluids. However, the
stability conditions of the homogeneous equilibrium of the second-order
Müller–Israel–Stewart theory, the inertial extension of the first order one,
do not follow from thermodynamic principles, the conditions are beyond
thermodynamics, [30]. For higher-order theories, there are no related
investigations.

In case of nonrelativistic dissipative fluids, the linear stability of the
homogeneous thermodynamic equilibrium is proved for the Fourier–Navier–Stokes
system, and, according to the expectations, the conditions are purely
thermodynamical [31]111The stability is valid for any thermodynamically stable
EOS with concave entropy density and with nonnegative viscosities and heat
conduction coefficient. Apparently, [31] is the first general proof of this
kind in the literature.. The question arises, what are the conditions for the
second- and higher-order theories of nonrelativistic continua?

Restricting ourselves to heat conduction has advantages in the investigation:
it is simpler, focuses to the problematic aspects and several heat conduction
models are real: they are experimentally verified and tested. The various
theories may have different origins, sometimes not connected to
thermodynamics, and theory and experiments are interacting. A good example is
the discovery of non-Fourier heat conduction in room temperature heterogeneous
materials [32, 33, 34]. Whether the homogeneous equilibrium of higher-order
heat conduction theories (e.g. in phonon hydrodynamics) is stable and whether
the stability conditions stretch beyond pure thermodynamics, like in special
relativistic dissipative fluids, is an open question.

Figure 1: The hierarchy of nonrelativistic and special relativistic fluids and
the role of thermodynamic conditions in the stability. For thermostatics it is
only the concave entropy, but for ordinary thermodynamics it is Lyapunov
stability. For continuum theories it is the linear stability of their
homogeneous thermodynamic equilibrium. MCV stands for
Maxwell–Cattaneo–Vernotte, and BDNK for Bemfica– Disconzi–Noronha–Kovtun

In general, non-Fourier heat conduction is a cornerstone and a test ground of
nonequilibrium thermodynamics. Several models exist [34], and the various
constitutive equations may have very different justifications. Thermodynamic
compatibility sometimes is not a requirement and this is a source of
unacceptable properties and instabilities [35, 36, 37]. Nonequilibrium
thermodynamics with internal variables united most of the different models in
a uniform framework, where the evolution equations for the heat flux (or a
vectorial internal variable in general [38]) are derived with a constructive
application of the Second Law of Thermodynamics [39], and preserved the
compatibility with Rational Extended Thermodynamics with notable differences
[40]. The universality of the thermodynamic framework is justified by the
theoretical prediction and experimental discovery of room temperature non-
Fourier heat conduction in heterogeneous materials [32, 33]. In this framework
the system of equations that lead to ballistic propagation is obtained, which
is a third-order theory according to the terminology of relativistic fluid
mechanics (because of the three coupled evolution equations).

In this paper, the linear stability of homogeneous equilibrium of third order
non-Fourier heat conduction in rigid heat conductors is analysed. The
investigation is restricted to the most general nonequilibrium thermodynamic
theory, which is directly compatible with kinetic theory of rarefied gases.
Therefore, the basic fields are the temperature, the heat flux and a tensorial
internal variable, and the corresponding theories will be called Extended
Theories of Heat Conduction because they generalise heat conduction in
Extended Thermodynamics. The general treatment with three spatial dimensions
seems to be straightforward, but it is somehow complicated even with the
simplest material symmetry in the isotropic case [41], therefore we restrict
ourselves to one spatial dimension, where the corresponding generalisations
can be easily classified and thermodynamic properties analysed.

The paper is organised as follows. In the next section the short outline of
the nonequilibrium thermodynamic background of the system of equations is
presented. Then the various forms of the constitutive evolution equations are
treated in one spacial dimension. In section three the stability analysis is
performed. In section four we treat various special theories and analyse the
role of thermodynamic conditions and stability of homogeneous equilibrium. Our
general conclusion is that thermodynamics ensures the linear stability of the
homogeneous equilibrium, but, surprisingly, the particular interpretation of
the internal variable seems to be too restrictive. The discussion and some
outlook are treated in the last section.

##  2 Third order Extended Heat Conduction – an outline of thermodynamic
background

In theories of heat conduction, the balance of internal energy is the starting
point. It can be written as

$\rho\dot{e}+\partial_{i}q^{i}=0,$ (1)

where the dot denotes the substantial (material) time derivative,
$\dot{e}=\partial_{t}e+v^{i}\partial_{i}e$, $\partial_{i}$ is the spatial
derivative, $\rho$ is the mass density, $e$ is the specific internal energy,
and $q^{i}$ is the heat flux, the conductive part of the current density of
the internal energy, where $i=1,2,3$. Double indices denote the usual dual
contraction, e.g. $\partial_{i}q^{i}=\nabla\cdot\bf q$.

Spacetime. Equation (1) is the usual substantial form of the fundamental
balance of energy conservation, where the spacetime derivative and the energy
four-vector are split into timelike and spacelike parts in a frame that is
comoving with the material [42]. The index notation is abstract; it has
nothing to do with coordinates. It is also remarkable that the upper indices
denote (contravariant) three-vectors and the lower ones three-covectors. The
balance (1) is a four-divergence of the internal energy four-vector from a
spacetime point of view.

The material properties are determined by the following way:

  1. 1.

The thermodynamic state space and the constitutive functions. Here the basic
difference between theories is whether the heat flux is considered as a
constitutive function as in case of Fourier theory and in internal variable
approaches or as a basic field for Extended Theories of Heat Conduction that
keep compatibility with Kinetic Theory [43, 38]. Here we will choose the
latter one, therefore $q^{i}$ is considered as a basic field and the evolution
equation of the heat flux will be determined constitutively. In this paper the
thermodynamic state space is spanned by the specific internal energy, $e$, the
heat flux, $q^{i}$, and by a second order three tensor field $Q^{ij}$, that
turns out to be the flux of the heat flux. Naturally one requires evolution
equations for both additional basic fields $q_{i}$ and $Q^{ij}$, respectively.

Objectivity. For constitutively generated evolution equations the requirement
of frame independence is a necessary condition of spacetime compatibility. For
evolution equations the objectivity of time derivatives is an evident
requirement. The simple substantial derivatives are not sufficient, other
objective time derivatives may give better theories. In mathematical models of
heat conducting moving media the various possibilities are not equivalent, the
well-posedness and of the related mathematical problems and stability of the
continuum theory is strongly connected to the form of objective derivatives
[44, 45]. According to the recent results of Angeles the choice of the
natural, Lie derivatives of the corresponding physical quantities is
preferable [46]. Thermodynamics may contribute to justify the proper choices
and remove the seemingly ad-hoc use of various objective forms [47].

  2. 2.

The second choice is the equation of state, which determines the entropy
density as a function of the thermodynamic state space variables. In the
classical Fourier theory it is the caloric equation of state given as $e=cT$,
where $T$ is the temperature and $c$ is the specific heat capacity. In our
case it will be given in an extended form, where the entropy depends on the
internal energy, $e$, the heat flux, $q^{i}$, and also on the the flux of the
heat flux, $Q^{ij}$, as follows

$s(e,q^{i},Q^{ij})=\tilde{s}\left(e-\frac{1}{2}q^{i}\widehat{m}_{ij}q^{j}-\frac{1}{2}Q^{ij}\widehat{M}_{jilk}Q^{kl}\right).$
(2)

The argument of $\tilde{s}$ will be denoted by
$u=e-\frac{1}{2}q^{i}\widehat{m}_{ij}q^{j}-\frac{1}{2}Q^{ij}\widehat{M}_{jilk}Q^{kl}$.
The quadratic form ensures the simplest functional realisation of
thermodynamic stability in all variables of the entropy function, together
with a simple interpretation. Therefore $\beta=1/T=\frac{d\tilde{s}}{du}$,
where the three variable function $s$ is distinguished from $\tilde{s}$, that
depends solely on $u$. Also, the thermodynamic inductivities
$\widehat{m}_{ij}$ and $\widehat{M}_{jilk}$ are introduced, which are
symmetric and positive definite, and, for the sake of convenience, are assumed
constant. If $\tilde{s}$ is concave, that is
$\frac{d^{2}\tilde{s}}{du^{2}}\leq 0$, then the three-variable entropy
function $s$ is concave, too, as it is seen calculating its negative definite
second derivative:

$D^{2}s=\begin{pmatrix}\frac{d^{2}\tilde{s}}{du^{2}}=-\frac{1}{T^{2}}\frac{dT}{du}&0&0\\\
0&-m_{ij}&0\\\ 0&0&-M_{ijkl}\end{pmatrix}.$ (3)

  3. 3.

The third important factor is the entropy flux, which is expected to be zero
if $q^{i}$ and $Q{ij}$ are zero, and therefore it is best represented as:

$J^{i}(e,q^{i},Q^{ij})=b^{i}_{j}q^{j}+B^{i}_{kj}Q^{jk}.$ (4)

Here the $b^{i}_{j}$ and $B^{i}_{kj}$ coefficient functions are the so-called
Nyíri multipliers, [48].

Fourier theory of heat conduction assumes that the heat flux is a constitutive
function. Previously the heat flux $q^{i}$, was considered a basic field.
Therefore, one is looking for the evolution equations of the basic fields and
the Nyíri multipliers are to be determined constitutively.

The constitutive functions determine the material properties, therefore they
are restricted by the Second Law of thermodynamics, the entropy balance:

$\displaystyle\rho\dot{s}+\partial_{i}J^{i}=\Sigma\geq 0.$ (5)

Considering (4) and (2) one can calculate the entropy production rate and
regroup the terms into the following convenient form:

$\displaystyle\Sigma=-q^{i}\left(m_{ij}\dot{q}^{j}-\partial_{j}b^{j}_{i}\right)+\left(b^{i}_{j}-\frac{1}{T}\delta^{i}_{j}\right)\partial_{i}q^{j}-Q^{ij}\left(M_{jilk}\dot{Q}^{kl}-\partial_{k}B^{k}_{ji}\right)+B^{i}_{kj}\partial_{i}Q^{jk}\geq
0,$ (6)

where $m_{ij}=\frac{\rho}{T}\hat{m}_{ij}$ and
$M_{ijkl}=\frac{\rho}{T}\hat{M}_{ijkl}$.

The constitutive functions are the Nyíri multipliers, $b^{i}_{j},B^{i}_{kj}$,
and the evolution equations, more properly, the right-hand side of the
following formulae: $\dot{q}^{i}=f^{i}$ and $\dot{Q}^{ij}=F^{ij}$. Then the
linear solution of the entropy inequality is the following:

$\displaystyle\begin{pmatrix}m_{ij}f^{j}-\partial_{j}b^{j}_{i}\\\
b^{j}_{k}-\frac{1}{T}\delta^{j}_{k}\\\
M_{lmvw}F^{wv}-\partial_{w}B^{w}_{lm}\\\
B^{n}_{op}\end{pmatrix}=\begin{pmatrix}{}^{11}L_{ia}&{}^{12}L_{ic}^{b}&{}^{13}L_{ide}&{}^{14}L_{igh}^{f}\\\
{}^{21}L^{j}_{ka}&{}^{22}L^{j\;\;b}_{\;kc}&{}^{23}L^{j}_{kde}&{}^{24}L^{j\;\;f}_{\;kgh}\\\
{}^{31}L_{lma}&{}^{32}L_{lmc}^{b}&{}^{33}L_{lmde}&{}^{34}L_{lmgh}^{f}\\\
{}^{41}L^{n}_{opa}&{}^{42}L^{n\;\;b}_{opc}&{}^{43}L^{n}_{opde}&{}^{44}L^{n\;\;\;f}_{opgh}\end{pmatrix}\begin{pmatrix}-q^{a}\\\
\partial_{b}q^{c}\\\ -Q^{ed}\\\ \partial_{f}Q^{hg}\end{pmatrix}.$ (7)

Here the symmetric part of the coefficient matrix is positive definite,
therefore the linear solution transforms the entropy inequality into a
positive definite quadratic form, where the transport matrices reflect the
symmetries of the continuum material. The simplest isotropic symmetry is
treated in detail in [41]. It leads to unexpected results, due to the third
order tensorial thermodynamic forces and fluxes. The cross-effects are
unexpected in the sense that they contradict the heuristic Curie principle:
the representation theorems of isotropic tensors show, that thermodynamic
interactions with different tensorial order may be coupled in isotropic
materials, too.

###  2.1 One-dimensional heat transport

In the following we restrict ourselves to one spatial dimension in Cartesian
coordinates, where

$\displaystyle B:=B^{1}_{11},\quad Q:=Q^{11},\quad q:=q^{1},\quad
b:=b^{1}_{1}.$ (8)

Also, the thermodynamic inductivities are reduced to single scalars
$m:=m_{11}$ and $M=M_{1111}$ in one dimension, independently of material
symmetries. Also according to thermodynamic stability,

$m>0,\qquad\text{and}\qquad M>0.$ (9)

In the one dimensional case the entropy flux (4) and the entropy production
(6) can be written respectively as

$\displaystyle J_{1}=bq+BQ,$ (10)
$\displaystyle\Sigma_{1}=-q\left(m\dot{q}-\partial_{x}b\right)+\left(b-\frac{1}{T}\right)\partial_{x}q-Q\left(M\dot{Q}-\partial_{x}B\right)+B\partial_{x}Q\geq
0.$ (11)

Here $\partial_{x}$ indicates the spatial derivative with respect to $x$. The
field quantities are assumed to change only along the direction of the $x$
coordinate.

The linear solution of the entropy inequality becomes

$\displaystyle
m\dot{q}-\partial_{x}b=-\lambda_{11}q+\lambda_{12}\partial_{x}Q,$ (12)
$\displaystyle b-\frac{1}{T}=\kappa_{11}\partial_{x}q-\kappa_{12}Q,$ (13)
$\displaystyle M\dot{Q}-\partial_{x}B=\kappa_{21}\partial_{x}q-\kappa_{22}Q,$
(14) $\displaystyle B=-\lambda_{21}q+\lambda_{22}\partial_{x}Q.$ (15)

The $\lambda$ coefficients indicate cross effects between vectors and third
order tensors in isotropic materials. That follows from the representation
theorems and are a counterexample of the Curie principle [41].

The system of constitutive equations is the same as (93)-(96) in [41], with
the following notation of the isotropic conductivity coefficients:

$\displaystyle\begin{pmatrix}\lambda_{11}&0&0&\lambda_{12}\\\
0&\kappa_{11}&\kappa_{12}&0\\\ 0&\kappa_{21}&\kappa_{22}&\\\
\lambda_{21}&0&0&\lambda_{22}\end{pmatrix}:=\begin{pmatrix}L^{(1)}&0&0&-L^{(1,4)}\\\
0&L^{(2)}&-L^{(2,3)}&0\\\ 0&-L^{(3,2)}&L^{(3)}&0\\\
-L^{(4,1)}&0&0&C^{(4)}\end{pmatrix}.$ (16)

The new notation on the left hand side is introduced, because the calculations
below and also the recent applications and models are one dimensional. The
differences in the signs are consistent with the flux-force system in our
paper (and mistaken in [41]), however, the conditions of nonnegative entropy
production are the same. It is also remarkable that in [41] the odd and even
parity of $Q^{ii}$ were analyzed separately. However, here we postpone any
further assumptions regarding time reversal symmetry, microscopic or
macroscopic irreversibility, [49], because the general approach will be
helpful in the stability analysis.

The balance form of (12) and (14) is remarkable: $-b$ and $-B$ are the current
densities.

Substituting the constitutive functions (12)-(15), one can obtain the one
dimensional entropy flux (10), and entropy production (11), as follows:

$\displaystyle
J_{1}=\frac{q}{T}+\kappa_{11}\partial_{x}\left(\frac{q^{2}}{2}\right)+\lambda_{22}\partial_{x}\left(\frac{Q^{2}}{2}\right)-\left(\kappa_{12}+\lambda_{21}\right)qQ,$
(17)
$\displaystyle\Sigma_{1}=\lambda_{11}q^{2}+\lambda_{22}(\partial_{x}Q)^{2}+(\lambda_{12}+\lambda_{21})(-q)\partial_{x}Q+\kappa_{11}(\partial_{x}q)^{2}+\kappa_{22}Q^{2}+$
(18) $\displaystyle(\kappa_{12}+\kappa_{21})(-Q)\partial_{x}q\geq 0.$

One can see from the quadratic form of the entropy production, that the
conductivity coefficients must fulfil the following inequalities:

$\displaystyle\lambda_{11}\geq 0,\quad\lambda_{22}\geq
0,\quad\lambda_{11}\lambda_{22}-\frac{(\lambda_{12}+\lambda_{21})^{2}}{4}\geq
0,$ (19) $\displaystyle\kappa_{11}\geq 0,\quad\kappa_{22}\geq
0,\quad\kappa_{11}\kappa_{22}-\frac{(\kappa_{12}+\kappa_{21})^{2}}{4}\geq 0.$
(20)

Let us introduce the symmetric and antisymmetric crosscoefficients with the
help of the following notation:

$\displaystyle\lambda:=\frac{\lambda_{12}+\lambda_{21}}{2},\quad\hat{\lambda}:=\frac{\lambda_{12}-\lambda_{21}}{2},\quad\kappa=\frac{\kappa_{12}+\kappa_{21}}{2},\quad\hat{\kappa}=\frac{\kappa_{12}-\kappa_{21}}{2}.$
(21)

Then the following inequalities follow:

$\displaystyle\Lambda:=\lambda_{11}\lambda_{22}-\lambda_{12}\lambda_{21}=\lambda_{11}\lambda_{22}-\lambda^{2}+\hat{\lambda}^{2}\geq
0,$ (22) $\displaystyle
K:=\kappa_{11}\kappa_{22}-\kappa_{12}\kappa_{21}=\kappa_{11}\kappa_{22}-\kappa^{2}+\hat{\kappa}^{2}\geq
0.$ (23)

Therefore, the antisymmetric parts of the matrices positively contribute to
the determinant.

Eliminating $b$ and $B$ from the system (12)-(15) one obtains the evolution
equations of $q$ and $Q$:

$\displaystyle
m\dot{q}+\lambda_{11}q-\kappa_{11}\partial_{xx}q+(\kappa_{12}-\lambda_{12})\partial_{x}Q-\partial_{x}\beta$
$\displaystyle=0,$ (24) $\displaystyle
M\dot{Q}+\kappa_{22}Q-\lambda_{22}\partial_{xx}Q+(\lambda_{21}-\kappa_{21})\partial_{x}q$
$\displaystyle=0,$ (25)

where $\beta=1/T$ is the reciprocal temperature.

Also, we have assumed, that the conductivity coefficients $\kappa_{11}$,
$\kappa_{12}$, $\lambda_{22}$ and $\lambda_{21}$ are constants. One can obtain
the rate equation of the heat flux $q$ by eliminating the internal variable
$Q$ as well:

$\displaystyle
mM\ddot{q}+(m\kappa_{22}+M\lambda_{11})\dot{q}+\kappa_{22}\lambda_{11}q-(m\lambda_{22}+M\kappa_{11})\partial_{xx}\dot{q}-$
(26)
$\displaystyle(\Lambda+K+\kappa_{12}\lambda_{21}+\kappa_{21}\lambda_{12})\partial_{xx}q+\kappa_{11}\lambda_{22}\partial_{xxxx}q=$
$\displaystyle
M\partial_{x}\dot{\beta}+\kappa_{22}\partial_{x}\beta-\lambda_{22}\partial_{xxx}\beta.$

If $\lambda_{11}\neq 0$ and $\kappa_{22}\neq 0$ then one can introduce a more
insightful notation for the coefficients:

$\displaystyle\tau_{Q}:=\frac{M}{\kappa_{22}},\quad\tau_{q}:=\frac{m}{\lambda_{11}},\quad
l_{Q}:=\left|\sqrt{\frac{\lambda_{22}}{\kappa_{22}}}\right|,\quad
l_{q}:=\left|\sqrt{\frac{\kappa_{11}}{\lambda_{11}}}\right|,\quad\lambda_{th}:=\frac{1}{\lambda_{11}},$
(27)

where $\tau_{Q}$ and $\tau_{q}$ are relaxation times, $l_{Q}$ and $l_{q}$ are
characteristic lengths and $\lambda_{th}$ is the thermodynamic heat conduction
coefficient. Then, the Fourier heat conduction coefficient can be written as
$\lambda_{F}=\frac{1}{T^{2}\lambda_{th}}$. The two versions of heat conduction
coefficients appear because the thermodynamic force of heat conduction is the
gradient of the reciprocal temperature and not the gradient of the
temperature. Furthermore, we introduce a third characteristic length as

$\displaystyle\hat{l}_{q}:=\left|\sqrt{l_{q}^{2}+a_{q}a_{Q}}\right|,\quad\text{where}\quad
a_{q}:=\frac{\kappa_{12}-\lambda_{12}}{\lambda_{11}},\quad\text{and}\quad
a_{Q}:=\frac{\lambda_{21}-\kappa_{21}}{\kappa_{22}}.$ (28)

Therefore, the system of equations (24)-(25) can be written as

$\displaystyle\tau_{q}\dot{q}+q-l_{q}^{2}\partial_{xx}q+a_{Q}\partial_{x}Q$
$\displaystyle=\lambda_{th}\partial_{x}\beta,$ (29)
$\displaystyle\tau_{Q}\dot{Q}+Q-l_{Q}^{2}\partial_{xx}Q+a_{q}\partial_{x}q$
$\displaystyle=0.$ (30)

Also the rate equation of the heat flux (26) can be rewritten in the following
form:

$\displaystyle\tau_{Q}\tau_{q}\ddot{q}+(\tau_{q}+\tau_{Q})\dot{q}+q-(\tau_{q}l_{Q}^{2}+\tau_{Q}l_{q}^{2})\partial_{xx}\dot{q}-(l_{Q}^{2}+\hat{l}_{q}^{2})\partial_{xx}q+l_{Q}^{2}l_{q}^{2}\partial_{xxxx}q-$
(31)
$\displaystyle\lambda_{th}\left(\tau_{Q}\partial_{x}\dot{\beta}+\partial_{x}\beta-
l_{Q}^{2}\partial_{xxx}\beta\right)=0.$

We can see that the number of relevant coefficients is reduced to two
relaxation times and three characteristic length scales beyond the heat
conduction coefficient. It is also important that the rate equation of the
heat flux can be reorganised into the following instructive form:

$\displaystyle(\tau_{Q}d_{t}-l_{Q}^{2}\partial_{xx})(\tau_{q}\dot{q}+q-l_{q}^{2}\partial_{xx}\dot{q}-\lambda_{th}\partial_{x}\beta)+(\tau_{q}\dot{q}+q-\hat{l}_{q}^{2}\partial_{xx}\dot{q}-\lambda_{th}\partial_{x}\beta)=0.$
(32)

Here $d_{t}$ appears as the time derivative operator ($d_{t}q=\dot{q}$). This
is the hierarchical form of the rate equation of the heat flux. The
hierarchical form is instructive, because it reflects the structure of the
system of evolution equations before the elimination of the second order
tensor component $Q$. Presumably, the above hierarchy is valid also in three
(or two) spatial dimensions and characterizes the constraint of higher order
heat conduction originated in the Second Law of Thermodynamics.

Finally, the balance of internal energy (1), can be written in one spatial
dimension as

$\rho c_{v}\dot{T}+\partial_{x}q=0,$ (33)

where

$c_{v}=\frac{de}{dT}>0,$ (34)

according to thermodynamic stability. This last equation completes the
previous constitutive equations.

##  3 Exponential plane waves

In this section, analogously as in [31], we consider the perturbations of the
physical quantities $T,q$ and $Q$ starting from their equilibrium values
$T_{0}\neq 0$, $q_{0}=0$ and $Q_{0}=0$, respectively. Therefore, we have

$T\approx T_{0}+\delta T,\qquad q\approx\delta q,\qquad Q\approx\delta Q.$
(35)

Then, from the system of perturbed equations (33), (24) and (25) we obtain

$\displaystyle\rho c_{v}\partial_{t}\delta T+\partial_{x}\delta q=0,$ (36)
$\displaystyle m\partial_{t}\delta q+\lambda_{11}\delta
q-\kappa_{11}\partial_{xx}\delta q+\frac{1}{T_{0}^{2}}\partial_{x}\delta
T+(\kappa_{12}-\lambda_{12})\partial_{x}\delta Q=0,$ (37) $\displaystyle
M\partial_{t}\delta Q+\kappa_{22}\delta Q-\lambda_{22}\partial_{xx}\delta
Q+(\lambda_{21}-\kappa_{21})\partial_{x}\delta q=0.$ (38)

We look for the exponential plane wave solutions of this system of equations,
i.e.

$\delta T=\widehat{\delta T}e^{\Gamma t+ikx},\qquad\delta q=\widehat{\delta
q}e^{\Gamma t+ikx},\qquad\delta Q=\widehat{\delta Q}e^{\Gamma t+ikx}.$ (39)

From these we obtain

$\displaystyle\rho c_{v}\Gamma{\delta T}+ik{\delta q}=0,$ (40) $\displaystyle
ik{\delta
T}+T^{2}_{0}\left(m\Gamma+\lambda_{11}+\kappa_{11}k^{2}\right){\delta
q}+T^{2}_{0}(\kappa_{12}-\lambda_{12})ik{\delta Q}=0,$ (41)
$\displaystyle(\lambda_{21}-\kappa_{21})ik{\delta
q}+\left(M\Gamma+\kappa_{22}+\lambda_{22}k^{2}\right){\delta Q}=0.$ (42)

The system of equations (40)-(42) has non-trivial solutions only if its
determinant vanishes, i.e.

$\begin{vmatrix}\rho c_{v}\Gamma&ik&0\\\\[5.0pt]
ik/T^{2}_{0}&\left(m\Gamma+\lambda_{11}+\kappa_{11}k^{2}\right)&(\kappa_{12}-\lambda_{12})ik\\\\[5.0pt]
0&(\lambda_{21}-\kappa_{21})ik&\left(M\Gamma+\kappa_{22}+\lambda_{22}k^{2}\right)\end{vmatrix}=0.$
(43)

From (43) we find the following polynomial:

$\begin{split}0&=mM\Gamma^{3}+\left[m\kappa_{22}+M\lambda_{11}+\left(m\lambda_{22}+M\kappa_{11}\right)k^{2}\right]\Gamma^{2}+\\\
&+\left[\frac{k^{2}M}{T^{2}_{0}\rho
c_{v}}+\lambda_{11}\kappa_{22}+\left(\Lambda+K+\lambda_{12}\kappa_{21}+\lambda_{21}\kappa_{12}\right)k^{2}+\lambda_{22}\kappa_{11}k^{4}\right]\Gamma+\\\
&+\frac{k^{2}}{T^{2}_{0}\rho
c_{v}}\left(\kappa_{22}+\lambda_{22}k^{2}\right).\end{split}$ (44)

To find the conditions of stability for the exponential plane waves (39) we
discuss the (44) dispersion relation. It has the form

$a_{0}\Gamma^{3}+a_{1}\Gamma^{2}+a_{2}\Gamma+a_{3}=0,$ (45)

where $a_{0},a_{1},a_{2}$ and $a_{3}$ are in equation (39), respectively, the
coefficients of $\Gamma^{3},\Gamma^{2},\Gamma$, and the last term. Then,
according to the Routh-Hurwitz criteria (see [50]), the real parts of the
solutions of (45) are negative if every coefficient is nonnegative and also

$\displaystyle 0$ $\displaystyle<a_{1}a_{2}-a_{0}a_{3}=$ (46)
$\displaystyle=\kappa_{11}\lambda_{22}\left(m\lambda_{22}+M\kappa_{11}\right)k^{6}+$
$\displaystyle+\left[\frac{M^{2}\kappa_{11}}{\rho
c_{v}T^{2}_{0}}+\kappa_{11}\lambda_{22}\left(m\kappa_{22}+M\lambda_{11}\right)+\right.$
$\displaystyle\qquad\left.+\left(m\lambda_{22}+M\kappa_{11}\right)\left(\underline{\Lambda+K+\lambda_{12}\kappa_{21}+\lambda_{21}\kappa_{12}}\right)\right]k^{4}+$
$\displaystyle+\left[\frac{M^{2}\lambda_{11}}{\rho
c_{v}T^{2}_{0}}+\lambda_{11}\kappa_{22}\left(m\lambda_{22}+M\kappa_{11}\right)+\right.$
$\displaystyle\qquad\left.+\left(m\kappa_{22}+M\lambda_{11}\right)\left(\underline{\Lambda+K+\lambda_{12}\kappa_{21}+\lambda_{21}\kappa_{12}}\right)\right]k^{2}+$
$\displaystyle+\lambda_{11}\kappa_{22}\left(m\kappa_{22}+M\lambda_{11}\right).$

One can see, that $a_{0}>0,a_{1}>0,a_{3}>0$, follow from pure thermodynamic
conditions, namely from thermodynamic stability (34), (9) and the
inequalities, (19) and (20), required by nonnegative entropy production.
However, the above inequality and the condition $a_{2}>0$ are fulfilled only
if

$\displaystyle\Lambda+K+\lambda_{12}\kappa_{21}+\lambda_{21}\kappa_{12}=$ (47)
$\displaystyle\lambda_{11}\lambda_{22}+\kappa_{11}\kappa_{22}-(\lambda-\kappa)^{2}+(\hat{\lambda}-\hat{\kappa})^{2}\geq
0,$

or, dividing the above inequality by $\lambda_{11}\kappa_{22}$ one obtains the
stability condition in terms of characteristic lengths

$\displaystyle l_{q}^{2}+l_{Q}^{2}+a_{q}a_{Q}\geq 0,$ (48)

Here we have introduced the notation stated in equation (21), for the
symmetric and antisymmetric crosscoefficients. It is apparent from the second
form of the inequality that the thermodynamic conditions for the conductivity
coefficients (22)-(23), ensure only that
$\lambda_{11}\lambda_{22}-\lambda^{2}+\kappa_{11}\kappa_{22}-\kappa^{2}+(\hat{\lambda}-\hat{\kappa})^{2}\geq
0$ and the possibility

$\displaystyle\lambda_{11}\lambda_{22}-\lambda^{2}+\kappa_{11}\kappa_{22}-\kappa^{2}+(\hat{\lambda}-\hat{\kappa})^{2}\leq-2\lambda\kappa$
(49)

does not violate the second law. Interestingly that condition requires
symmetric cross-coefficients, $\lambda$ and $\kappa$, that contribute to the
dissipative part of the entropy production. That cannot appear in the known,
hyperbolic framework of Extended Thermodynamics. We will see that in the next
section, where the special cases of the general framework will be analysed.

##  4 Special cases

The special cases are best analysed with conditions for the original
thermodynamic coefficients, because those represent the proper material
properties and the most general form of the second law restrictions. However,
the physical parameters provide a simpler interpretation and insight,
therefore we first impose the conditions in physical parameters, then extend
the analysis to the cases that are not tractable with them.

###  4.1 Decoupled heat conduction: $a_{q}=0$ or $a_{Q}=0$

It is easy to see, that the previous general system of constitutive partial
differential equations (29)-(30) becomes decoupled if either $a_{q}=0$ or
$a_{Q}=0$, therefore $l:=l_{q}=\hat{l}_{q}$. If both conditions are fulfilled
then the two equations are independent of each other. However, if only one of
the coupling parameters is zero, then the solution of the decoupled equation
appears as a source term in the other side. In particular if $a_{Q}=0$, then

$\displaystyle\tau_{q}\dot{q}+q-l_{q}^{2}\partial_{xx}q+a_{q}\partial_{x}Q$
$\displaystyle=\lambda_{th}\partial_{x}\beta,$ (50)
$\displaystyle\tau_{Q}\dot{Q}+Q-l_{Q}^{2}\partial_{xx}Q$ $\displaystyle=0.$
(51)

Therefore, the solution of (51) – a partial differential equation reminiscent
of the Fourier one with an additional heat exchange source term – provides a
source term of the MCV type evolution equation of the heat flux according to
(50). This simple physical insight is not apparent neither from the decoupled
rate form

$\displaystyle\tau_{Q}\tau_{q}\ddot{q}+(\tau_{q}+\tau_{Q})\dot{q}+q-(\tau_{q}l_{Q}^{2}+\tau_{Q}l_{q}^{2})\partial_{xx}\dot{q}-(l_{Q}^{2}+l_{q}^{2})\partial_{xx}q+l_{Q}^{2}l_{q}^{2}\partial_{xxxx}q-$
(52)
$\displaystyle\lambda_{th}\left(\tau_{Q}\partial_{x}\dot{\beta}+\partial_{x}\beta-
l_{Q}^{2}\partial_{xxx}\beta\right)=0,$

nor from the simpler decoupled hierarchical form:

$\displaystyle(\tau_{Q}d_{t}+1-l_{Q}^{2}\partial_{xx})(\tau_{q}\dot{q}+q-l_{q}^{2}\partial_{xx}\dot{q}-\lambda_{th}\partial_{x}\beta)=0.$
(53)

The comparison of the differential operators of the hierarchical (53), and the
decoupled system forms (50)-(51) of heat conduction equations is remarkable
regarding the interpretation of any hierarchical arrangements of the evolution
equations.

It is straightforward to check that in both cases the inequality (48) is
fulfilled, because $l_{q}^{2}+l_{Q}^{2}\geq 0$. Therefore, the homogeneous
equilibrium of decoupled heat conduction is naturally stable, the linear
stability is the consequence of the Second Law of Thermodynamics.

####  4.1.1 Fourier, Maxwell–Cattaneo–Vernotte and Guyer–Krumhansl: $a_{q}=0$

The simplest heat conduction theories are obtained if (29) is independent of
(30). Then the Fourier heat conduction law emerges if both the relaxation time
$\tau_{q}$ and the characteristic length $l_{q}$ are zero, too. In this case
(29) is reduced to

$q=\lambda_{th}\partial_{x}\beta=-\lambda_{F}\partial_{x}T,$ (54)

where $\lambda_{F}=\lambda_{th}/T^{2}$ is the Fourier heat conduction
coefficient.

Also, if the relaxation time and the characteristic length are nonzero, one
can recover the Maxwell–Cattaneo–Vernotte equation

$\tau_{q}\dot{q}+q+\lambda_{F}\partial_{x}T=0,$ (55)

and the Guyer–Krumhansl equation, respectively

$\tau_{q}\dot{q}+q-l^{2}\partial_{xx}q+\lambda_{F}\partial_{x}T=0.$ (56)

Both (55) and (56) are differential equations, and together with the energy
balance (1) require proper initial and boundary conditions. Natural boundary
conditions can be understood by second law analysis, as it was researched e.g.
in [24, 51].

The stability polynomial of the Guyer–Krumhansl case is

$\tau_{q}\Gamma^{2}+\left(1+l^{2}k^{2}\right)\Gamma+\frac{\lambda_{F}}{\rho
c_{v}}k^{2}=0.$ (57)

Therefore, one can check directly that the linear stability of homogeneous
equilibrium is natural, in the sense that follows from the thermodynamic
conditions.

###  4.2 Extended Thermodynamics: $l_{q}=0$ and $l_{Q}=0$

In this case the second spatial derivatives in (29) and in (30) are zero, and
therefore, the system becomes analogous to the one dimensional version of the
9 field equations of Rational Extended Thermodynamics (see [10] page 351, eq.
(3.11)), and Extended Irreversible Thermodynamics, therefore can be compared
directly to heat conduction theories that are compatible with kinetic theory.
Let us remark that a derivation that is based on a particular microscopic
interpretation (e.g. rarefied monoatomic or polyatomic gases) cannot be
considered general. The obtained system of equations is referred to as
ballistic-conductive in [52].

The system of equations has the following form

$\displaystyle\tau_{q}\dot{q}+q+a\partial_{x}Q$
$\displaystyle=\lambda_{th}\partial_{x}\beta,$ (58)
$\displaystyle\tau_{Q}\dot{Q}+Q+a\partial_{x}q$ $\displaystyle=0.$ (59)

Here it was already considered that nonnegative entropy production requires
the coupling coefficients to be equal:

$a:=a_{q}=a_{Q},$ (60)

because the zero characteristic lengths $l_{q}=l_{Q}=0$ stem from the
conditions $\lambda_{22}=0$ and $\kappa_{11}=0$ for the original
thermodynamics coefficients, therefore the third inequalities of (19) and (20)
demand that the conductivity matrix is antisymmetric. This is also observed by
Rukolaine [53].

Then the rate equation of the heat flux, (31), for the Extended Thermodynamics
case becomes

$\displaystyle\tau_{Q}\tau_{q}\ddot{q}+(\tau_{q}+\tau_{Q})\dot{q}+q-a^{2}\partial_{xx}q-\lambda_{th}\left(\tau_{Q}\partial_{x}\dot{\beta}+\partial_{x}\beta\right)=0.$
(61)

The linear stability of homogeneous equilibrium of heat conduction in Extended
Thermodynamics follows from thermodynamic conditions, because the inequality
(48) is fulfilled as $a^{2}\geq 0$.

Materials that obey the above heat conduction equation have the interesting
property, that the initial internal energy is locally accumulated, therefore
it has a particular heat storage capability, independently of the heat
capacity of the material. This property is due to the hyperbolicity, the lack
of diffusive terms, that is the second order spatial derivatives in the system
of equations. The local accumulation of internal energy may be considered
unphysical as was observed recently by Rukolaine [53, 54]. It is also a
characteristic property of the Jeffreys equation [55].

It is also remarkable that if the coupling coefficients $a_{q}$ and $a_{Q}$
are connected to kinetic theory calculations then they are not necessarily
equal. For example in [10] $a_{q}=c^{2}\tau_{R}$ and
$a_{Q}=\frac{4}{15}\frac{\tau_{R}\tau_{N}}{\tau_{R}+\tau_{N}}$, where $c$ is
the Debye speed, $\tau_{R}$ and $\tau_{N}$ are the relaxation times of the
energy conserving resistive R-processes and energy and momentum conserving
normal N-processes of the Callaway model (see also the comparative study in
[40]). In the recent thermodynamic framework difference between $a_{q}$ and
$a_{Q}$ is possible only if at least one of the characteristic lengths are not
zero. However, such a diffusion terms cannot appear in Extended
Thermodynamics.

####  4.2.1 Burgers form: $l_{q}=0$, $l_{Q}=0$ and $a=0$

If $a=0$ in (61), then it looks like a Burgers equation

$\displaystyle\tau_{Q}\tau_{q}\ddot{q}+(\tau_{q}+\tau_{Q})\dot{q}+q-\lambda_{th}\left(\tau_{Q}\partial_{x}\dot{\beta}+\partial_{x}\beta\right)=0.$
(62)

However, a general Burgers equation with the reciprocal temperature looks like

$\displaystyle
b_{1}\ddot{q}+b_{2}\dot{q}+q+b_{3}\partial_{x}\dot{\beta}-\lambda_{th}\partial_{x}\beta=0.$
(63)

with four independent coefficients $b_{1},b_{2},b_{3}$ and $\lambda_{th}$. In
(62) the four coefficients are expressed by three parameters, $\tau_{q}$,
$\tau_{Q}$ and $\lambda_{th}$, therefore the Burgers equation is degenerated.
It is evident if it is rearranged in a hierarchical form, just like in (53),
as

$\displaystyle(\tau_{Q}d_{t}+1)(\tau_{q}\dot{q}+q-\lambda_{th}\partial_{x}\beta)=0,$
(64)

or remember that the system is decoupled and the rate equation of heat
conduction is actually a Maxwell–Cattaneo–Vernotte one.

####  4.2.2 Jeffreys equation: $l_{q}=l_{Q}=0$, $a=0$ and $\tau_{q}=0$

A general Jeffreys type equation in term of the reciprocal temperature $\beta$
appears if $b_{1}=0$ in the general Burgers equation, (63), and can be given
in the following form.

$\displaystyle
b_{2}\dot{q}+q+b_{3}\partial_{x}\dot{\beta}-\lambda_{th}\partial_{x}\beta=0.$
(65)

If the relaxation time $\tau_{q}$ is zero in (62), one obtains, that

$\displaystyle\tau_{Q}\dot{q}+q-\lambda_{th}\tau_{Q}\partial_{x}\dot{\beta}-\lambda_{th}\partial_{x}\beta=0.$
(66)

That looks like a Jeffreys type heat conduction. However, again there are
three coefficients and only two independent parameters and the hierarchical
form

$\displaystyle(\tau_{Q}d_{t}+1)(q-\lambda_{th}\partial_{x}\beta)=0,$ (67)

or the decoupled original system, (58)-(59), shows that it is a Fourier
equation for the heat flux $q$.

Therefore, neither the general Burgers nor the general Jeffreys equation
cannot be derived with the recent conditions. That fact remained hidden in
[52]. Also the degenerate Burgers and Jeffreys equations are linearly stable,
but thermodynamics of Extended Heat Conduction is silent about the stability
of the general ones. One may wonder what the reason of the unexpected
degeneracy of the evolution equations is…

###  4.3 Heat conduction without Fourier, $\lambda_{11}=0$

A particular family of general heat conduction equations does not contain
Fourier type heat conduction as a special case. For example the for the
Green–Naghdi II equations the heat flux part is simply missing, moreover, the
dissipation is zero. Such a possibility is not apparent in the previous
investigations because it is connected to singularities of the physical
coefficients, (27). However, considering the original thermodynamic
conductivity parameters one may recognize, that nothing excludes, that
$\lambda_{11}$, the coefficient of the heat flux in (13), be zero.

Then the system of equations (24)-(25) becomes

$\displaystyle\dot{q}-\xi_{q}\partial_{xx}q+b_{q}\partial_{x}Q-\kappa\partial_{x}\beta$
$\displaystyle=0,$ (68)
$\displaystyle\tau_{Q}\dot{Q}+Q-l^{2}_{Q}\partial_{xx}Q+a_{Q}\partial_{x}q$
$\displaystyle=0.$ (69)

Here the physical parameters in the second equation remained the same as
before, but the following coefficients were introduced in the first one:

$\xi_{q}:=\frac{\kappa_{11}}{m},\quad
b_{q}:=\frac{\kappa_{12}-\hat{\lambda}}{m},\quad\kappa:=\frac{1}{m}.$

Nonnegative entropy production with $\lambda_{11}=0$ requires, that the
related offdiagonal elements have an opposite sign, therefore
$\hat{\lambda}=\lambda_{12}=-\lambda_{21}$.

Now, the rate form can be written as

$\displaystyle\tau_{Q}\ddot{q}+\dot{q}-(l_{Q}^{2}+\tau_{Q}\xi_{q})\partial_{xx}\dot{q}-(\xi_{q}+a_{Q}b_{q})\partial_{xx}q+\xi_{q}l^{2}_{Q}\partial_{xxxx}q=$
(70) $\displaystyle\kappa(\tau_{Q}\partial_{x}\dot{\beta}+\partial_{x}\beta-
l_{Q}^{2}\partial_{xxx}\beta).$

The hierarchical form is

$\displaystyle(\tau_{Q}d_{t}-l_{Q}^{2}\partial_{xx})(\dot{q}-\xi_{q}\partial_{xx}{q}-\kappa\partial_{x}\beta)+(\dot{q}-\hat{\xi}_{q}\partial_{xx}{q}-\kappa\partial_{x}\beta)=0.$
(71)

Here $\hat{\xi}_{q}=\xi_{q}+a_{Q}b_{q}$. Previously, in (53), the hierarchical
structure revealed two Guyer–Krumhansl equations with different characteristic
lengths. Now, the heat flux term is missing and instead of Guyer–Krumhansl
equations one can observe two equations in the form
$\dot{q}-\xi\partial_{xx}{q}-\kappa\partial_{x}\beta=0$. Those are second type
Green–Naghdi equations if the the $\xi$ parameter is zero [56].

The inequality condition for the linear stability of the homogeneous
equilibrium, (72), in our case is written as

$\displaystyle\hat{\lambda}^{2}+K+\hat{\lambda}(\kappa_{21}-\kappa_{12})=\kappa_{11}\kappa_{22}-\kappa^{2}+(\hat{\lambda}-\hat{\kappa})^{2}\geq
0,$ (72)

Therefore, according to (20), the linear stability of homogeneous equilibrium
is warranted by thermodynamic conditions.

####  4.3.1 General Quintanilla: $\xi_{q}=0$ and $l_{Q}^{2}=0$

In this case (70) simplifies to

$\displaystyle\tau_{Q}\ddot{q}+\dot{q}-a_{Q}b_{q}\partial_{xx}q-\kappa(\tau_{Q}\partial_{x}\dot{\beta}+\partial_{x}\beta)=0.$
(73)

If the system is decoupled, then $a_{Q}=0$ and (68) is reduced to the
Quintanilla equation [57],

$\displaystyle\tau_{Q}\ddot{q}+\dot{q}-\kappa(\tau_{Q}\partial_{x}\dot{\beta}+\partial_{x}\beta)=0.$
(74)

This is a particular form, like in the case of Jeffreys equation, with two
independent coefficients only. The hierarchical rearrangement also reflects
the decoupling

$\displaystyle(\tau_{Q}d_{t}+1)(\dot{q}+\kappa\partial_{x}\beta)=0.$ (75)

Based on the linear stability analysis calculations in Section 3, the
stability polynomial of (74) is the following:

$\tau_{Q}\Gamma^{3}+\Gamma^{2}+n\tau_{Q}\Gamma+n=0,$ (76)

with the notation $n:=k^{2}\kappa/\rho c_{v}$.

Since the following condition can be obtained from the stability polynomial:

$n\tau_{Q}-n\tau_{Q}>0,$ (77)

the homogeneous equilibrium of the Quintanilla equation is marginally stable.
This means, that according to the applied Routh–Hurwitz criteria from (46),
eigenvalues of the Jacobian matrix have real parts.

##  5 Summary and discussion

In this paper a thermodynamic compatibility and stability analysis of third
order non-relativistic Extended Theories of Heat Conduction have been
developed. Several known special cases were investigated in the light of a
general approach of nonequilibrium thermodynamics with internal variables
(NeTIV) (see figure 2.) In this approach, the deviation from thermodynamic
equilibrium is characterised by additional fields, called internal variables.
The evolution equation of the internal variables is derived taking into
account the requirements of the Second Law of Thermodynamics: thermodynamic
stability (concave entropy) and non-negative entropy production.

Furthermore, the following modeling assumptions are necessary for
compatibility with the moment series expansion of kinetic theory and therefore
are characteristic in both Extended Irreversible Thermodynamics and Rational
Extended Thermodynamics:

  1. 1.

Increasing tensorial order of the basic fields. Therefore, two internal
variables were introduced in this work: a vectorial and a tensorial one.

  2. 2.

The vectorial internal variable is the heat flux.

  3. 3.

Both the vectorial and the tensorial internal variables vanish in
thermodynamic equilibrium when the entropy density depends only on the
internal energy and the entropy flux is the heat flux over the temperature.

As a consequence of the third assumption the entropy is a quadratic function
of the internal variables [5, 8], and the entropy flux is proportional to the
internal variables [48]. These properties do not require a particular
microstructure, like rarefied gases, they are general. Therefore, the
universality of the consequent heat conduction theories, like Fourier theory,
follows.

The investigations are restricted to isotropic materials and to one spatial
dimension. By eliminating the tensorial internal variable, the rate equations
of the heat flux were obtained, so that the deviation from thermodynamic
equilibrium was transformed into a deviation from the Fourier law, the usual
framework of non-Fourier heat conduction theories.

Figure 2: The relation of the structure of the evolution equations of the
various heat conduction theories in one spatial dimension, according to the
thermodynamic background. Several know heat conduction theories are related
through nonequilibrium thermodynamics. Also there are some theories and
concepts in heat conduction modelling, that are not incorporated in this
framework, like the Jeffreys type equation.

There are two main implications of our research:

  * •

The thermodynamic conditions – concave entropy and non-negative entropy
production – ensure the linear stability of homogeneous equilibrium in most
cases. However, the possible violation appears on the most general level. The
special cases, like theories of Extended Thermodynamics, are stable, because
the instability is connected to nonzero characteristic lengths of the
diffusion of the heat flux, $q^{i}$, and the diffusion of the flux of the heat
flux, $Q^{ij}$, and those terms are absent in the usual approaches.

  * •

One can obtain evolution equations of many known heat conduction theories in
the presented uniform theoretical framework of nonequilibrium thermodynamics.
However, in several important cases the derived equations were degenerate. For
example, in the derived Jeffreys-type equation the 3 material coefficients
were not independent. A similar situation was observed for the 9-field theory
of Extended Thermodynamics, for the Burgers equation and also for the
Quintanilla equation.

Regarding the first consequence, it should be noted that in the absence of a
microscopic background, we have not investigated Onsager or Casimir type
symmetries. However, if one considers that the heat flux $q$ is odd by time
reversal and the second order tensor $Q$ is even (that follows e.g. from
kinetic theory interpretation), then Casimir symmetry is valid for the cross-
coefficients, hence $\lambda_{12}=-\lambda_{21}$ and
$\kappa_{12}=-\kappa_{21}$. Therefore, the stability condition (72) is
satisfied and the linear stability of the homogeneous equilibrium follows from
the second law. However, this argument is based on the concept of macroscopic
reversibility, [49], and requires further analysis, in particular considering
some new developments in this respect, like [58, 59].

With regard to the second consequence, it can be noted that the reason why the
most general equations of heat conduction could not be obtained is apparently
due to the special choice of the basic fields. In fact, the deviation from
equilibrium was characterised by a vectorial internal variable with a fixed
physical interpretation: it was the heat flux according to our second
assumption. If the heat flux is not a basic field, but a constitutive
quantity, in the sense of Fourier heat conduction, then one can expect more
general heat conduction theories, with independent coefficients. For example,
in case of a single vectorial internal variable one can obtain the Jeffreys
equation without restrictions, [38, 60, 34]. According to this conjecture,
removing the restrictive interpretation of the vectorial internal variable
being the heat flux, there is a hope that one could understand the relation of
existing heat conduction theories within the single, uniform and constructive
framework of NeTIV.

A final remark seems to be suitable at the end. The form of the equations in
itself is not the whole theory. First of all, our analysis is restricted to
one spatial dimension, the various named theories are distinctly different in
a complete spacetime compatible framework. That complete framework defines the
physical parameters, the mathematical problems and also determines the
experimental approach. The physical parameters are comprehended differently in
kinetic theory motivated approaches and in nonequilibrium thermodynamics. The
boundary conditions and the structure of the equations determines the well or
ill posedness of the related mathematical problems. Finally, the solution of
the equations and the notions of the theory (e.g. universal or microstructure
dependent) guide the design of the experiments. Therefore, the outlined
connections of Figure 2 should be treated as guiding analogies.

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