Rep. Prog. Phys. 62 (1999) 1035–1142. Printed in the UK
PII: S0034-4885(99)91198-3
Extended irreversible thermodynamics revisited (1988–98)
D Jou†‡, J Casas-Vázquez† and G Lebon§
† Departament de Fı́sica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain
‡ Institut d’Estudis Catalans, Carme 47, 08001 Barcelona, Catalonia, Spain
§ Thermomécanique des Phénomènes Irreversibles, Université de Liège, Sart-Tilman B5, B-4000 Liège and Unitè
TERM, Université de Louvain, B-1348 Louvain-la-Neuve, Belgium
Received 19 October 1998, in final form 1 February 1999
Abstract
We review the progress made in extended irreversible thermodynamics during the ten years
that have elapsed since the publication of our first review on the same subject (Rep. Prog.
Phys. 1988 51 1105–72). During this decade much effort has been devoted to achieving a
better understanding of the fundamentals and a broadening of the domain of applications. The
macroscopic formulation of extended irreversible thermodynamics is reviewed and compared
with other non-equilibrium thermodynamic theories. The foundations of EIT are discussed
on the bases of information theory, kinetic theory, stochastic phenomena and computer
simulations. Several significant applications are presented, some of them of considerable
practical interest (non-classical heat transport, polymer solutions, non-Fickian diffusion,
microelectronic devices, dielectric relaxation), and some others of special theoretical appeal
(superfluids, nuclear collisions, cosmology). We also outline some basic problems which are
not yet completely solved, such as the definitions of entropy and temperature out of equilibrium,
the selection of the relevant variables, and the status to be reserved to the H -theorem and its
relation to the second law. In writing this review, we had four objectives in mind: to show (i)
that extended irreversible thermodynamics stands at the frontiers of modern thermodynamics;
(ii) that it opens the way to new and useful applications; (iii) that much progress has been
achieved during the last decade, and (iv) that the subject is far from being exhausted.
0034-4885/99/071035+108$90.00
© 1999 IOP Publishing Ltd
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Contents
1. Introduction
A. Macroscopic theory
2. EIT: motivations and general structure
2.1. Motivations
2.2. Generalized entropy and entropy flux in a one-component fluid
2.3. Entropy production and evolution equations for the fluxes
2.4. Rational extended thermodynamics
3. Non-equilibrium equations of state
3.1. Physical interpretation of the non-equilibrium entropy
3.2. Non-equilibrium equations of state for temperature, pressure and chemical
potential
3.3. Stability conditions
4. Comparison with other non-equilibrium theories
4.1. Theories with internal variables
4.2. Theories with fluctuations
5. Hamiltonian and variational formulations
5.1. Hamiltonian structure of classical thermo-hydrodynamics
5.2. Hamiltonian structure of extended irreversible thermodynamics
5.3. Variational formulations
B. Microscopic foundations
6. Information theory
6.1. Ideal gas under heat flux and viscous pressure: linear approximation
6.2. Ideal gas under shear flow: nonlinear analysis
6.3. Ideal gas with non-vanishing heat flux: nonlinear analysis
6.4. Ideal relativistic gas with non-vanishing heat flux
6.5. Harmonic chain with non-vanishing heat flux
7. Kinetic theory
7.1. Thirteen-moment approach
7.2. Higher-order moments
7.3. Infinite number of moments
7.4. Comparison with other results
7.5. Non-equilibrium entropy, the H -theorem and the second law
8. Stochastic processes
8.1. Persistent random walk
8.2. H -theorem for telegrapher-type equations
9. Computer simulations
C. Applications
10. Non-classical heat transport
10.1. A generalized Guyer–Krumhansl model
10.2. Onsager relations
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10.3. Heat propagation velocity
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10.4. Comparison with experimental results
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10.5. A generalized minimum entropy production principle
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10.6. Other topics
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11. Polymer solutions
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11.1. Macroscopic analysis
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11.2. Microscopic foundations
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11.3. Phase diagrams under shear flow
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12. Non-Fickian diffusion
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12.1. Case-II diffusion
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12.2. Taylor dispersion
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13. Dielectric relaxation of polar liquids
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14. Microelectronic devices
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14.1. Hydrodynamic description of transport
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14.2. Nonlinear effects. Flux limiters
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15. Superfluids
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16. Nuclear collisions
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16.1. Generalized Gibbs equation for nuclear matter
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16.2. Non-equilibrium corrections to the nuclear compressibility
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17. Viscous cosmological models
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17.1. Bulk viscosity and the evolution of the Universe
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17.2. Other topics: particle production, time-dependent cosmological constant,
astrophysical problems
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18. General conclusions
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Acknowledgments
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References
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1. Introduction
Extended irreversible thermodynamics (EIT), together with classical irreversible thermodynamics (CIT) and rational thermodynamics (RT) has been among the mainstream of research
in non-equilibrium thermodynamics. EIT crosses the borders of the local-equilibrium hypothesis and explores new grounds. Since the early 1960s, several hundred papers have been
published on EIT. Recently, two books (Jou et al 1993b, 1996a, Müller and Ruggeri 1993)
have appeared wherein a systematic analysis of EIT can be found. Other books closely related
to EIT but with different aims and scopes have also been published (Sieniutycz and Salomon
1992, Eu 1992, Sieniutycz 1994, Tzou 1997, Wilmanski 1998). In 1988, two review articles
(Jou et al 1988a, Garcı́a-Colı́n 1988) opened a wider perspective on EIT. Because of the intense
activity developed since the publication of our first review, we feel it necessary to update our
vision and to provide an overview of the progress achieved during the last decade. This review
has two objectives in mind: (1) to provide a deeper insight into the basic ideas underlying the
theory and (2) to stress its wide range of applications.
Motivated by the progress in material science and technology in general, the study of highfrequency, short-wavelength phenomena and (or) large amplitude perturbations has known
an extraordinary impetus. During the 1960s, experiments on light and neutron scattering
opened new perspectives and contributed to the development of the so-called generalized
hydrodynamics, where the usual transport laws are generalized to include memory and nonlocal effects. In the last two decades, the increasing miniaturization of physical devices (mainly
in microelectronics) has directed attention to a regime where transport is no longer dominated
by collisions amongst the particles, but rather by ballistic effects. All these features have
motivated the formulation of a mesoscopic description, intermediate between the macroscopic
and the microscopic ones. The main idea behind a statistical formalism is to eliminate the excess
of information resulting from a purely microscopic description. An alternative, complementary
attitude consists of starting from a macroscopic description and adding relevant information
to take into account peculiar features of the mesoscopic regime. The aim of EIT is precisely
the latter one, i.e. to propose an extension of classical thermodynamics towards mesoscopic
regimes. Obviously, the relevance of the theory should be checked by comparing it with
experiment and the results derived from microscopic models.
By EIT are understood the thermodynamic theories which use as independent variables
the dissipative fluxes in addition to the classical variables such as internal energy, density, mass
concentrations, deformation tensor etc. In the first versions of EIT, the extra variables were
selected to be the usual thermo-hydrodynamic fluxes as the heat flux, the viscous pressure
tensor, the electric flux, and the diffusion flux. Later, higher-order fluxes, such as the flux of
the heat flux, were included. Other keystones in the construction of EIT are the formulation
of evolution equations for the fluxes and the establishment of a generalized Gibbs equation
expressing the dependence of the non-equilibrium entropy with respect to the basic variables.
These lines of thought define a general framework, in which coexist several different methods
of approach, with either physical or mathematical emphasis. Seen from outside, this diversity
of points of view may be a source of confusion. Yet for people more familiar with the topic this
diversity has played a stimulating role, and has fostered efforts towards a unified view. Many
problems remain open in non-equilibrium thermodynamics in general and in EIT in particular.
It is, however, expected that the combination of an EIT approach and microscopic theories
will emerge into a suitable framework for the description of mesoscopic phenomena out of
equilibrium.
Section A contains a review of the basic ideas underlying EIT, followed by a comparison
with other theories of non-equilibrium thermodynamics. Section B asserts the microscopic
Extended irreversible thermodynamics revisited (1988–98)
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foundations of the theory, which are supported by the information theory, the kinetic theory,
the theory of stochastic processes and computer simulations. Section C is devoted to some
selected applications: some of special practical interest, such as non-classical heat transport,
polymer solutions, non-Fickian diffusion, dielectric relaxation, and microelectronic devices,
and also some other more particular subjects such as superfluids, nuclear collisions and viscous
cosmological models. Some sections are more technical and may be skipped in a first reading:
sections 5, 8, 13, 16 and 17.
It should be clearly stated that the present review is not a summary of the book on EIT (Jou
et al 1996a) published recently by the authors. Here we discuss more deeply some aspects
which were not treated in detail in the book; moreover, new topics are also developed. Our
objective is to be pedagogical and illustrative rather than exhaustive. Therefore, we have
concentrated our attention on some selected topics which we think are particularly suitable for
outlining the main physical aspects, and we have selected for bibliographical purposes only
the more significant papers. The reader interested in wider reference lists is advised to consult
our bibliographical surveys (Jou et al 1992, 1996b, 1998) as well as our database accessible
in the website http://circe.uab.es/eit.
A. Macroscopic theory
In the next sections we present the motivations and the essential ideas underlying EIT. Section 2
provides the generalized Gibbs equation and the evolution equations for the fluxes whose
possible forms are restricted by the second law. The generalized Gibbs equation, which
establishes a link between the dynamical equations for the fluxes and the equations of state, is
of special interest in the nonlinear regime. In section 3 we propose a physical interpretation of
the generalized entropy and of equations of state. In sections 4 and 5, EIT is compared with
other non-equilibrium approaches, such as internal variables theory, the theory of fluctuations
and Hamiltonian formulations.
2. EIT: motivations and general structure
2.1. Motivations
The first question to be asked in a thermodynamic theory is how to select the fundamental
variables. In its original formulation, EIT included the usual dissipative fluxes—heat flux,
diffusion flux, electric flux, viscous pressure—among the set of independent variables. The
main motivation was the need to describe phenomena with timescales comparable to the
relaxation times of these fluxes. Such situations are met when the phenomena are very
fast or very steep (as ultrasound propagation, light scattering in gases, neutron scattering
in liquids, heat propagation at low temperatures, shock waves etc) or when the relaxation
times of the fluxes become very long, as in polymeric solutions, suspensions, superfluids or
superconductors.
The simplest example justifying the choice of a flux as independent variable is found in
the problem of heat conduction in rigid bodies. The best known model for heat conduction
in undeformable solids is undoubtedly Fourier’s law which relates linearly the temperature
gradient ∇T (the cause) to the heat flux q (the effect)
q = −λ∇T
(2.1)
where λ is the heat conductivity, depending generally on the temperature. By combining (2.1)
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with the energy balance
∂T
ρcv
(2.2)
= −∇ · q
∂t
where ρ is the mass density and cv the heat capacity per unit mass at constant volume, one
obtains a parabolic differential equation for the temperature given by
∂T
ρcv
(2.3)
= ∇ · (λ∇T ).
∂t
Such an expression suffers from some pathological deficiencies, as pointed out by several
people and in particular by Onsager himself in his celebrated paper of 1931, where he noted
that Fourier’s model is in contradiction with the principle of microscopic reversibility. As
Onsager put it, this contradiction ‘. . . is removed when we recognize that [the Fourier law] is
only an approximate description of the process of conduction, neglecting the time needed for
acceleration of the heat flow’. In other words, Fourier’s law has the unphysical property that it
lacks inertial effects: if a sudden temperature perturbation is applied at one point in the solid,
it will be felt instantaneously and everywhere at distant points. Moreover, Fourier’s model
is not adequate for describing heat transport at very high frequencies and short wavelengths.
To eliminate this anomaly, Cattaneo (1948) proposed a damped version of Fourier’s law by
introducing a heat-flux relaxation term, namely,
∂q
τ
= −(q + λ∇T ).
(2.4)
∂t
Note that when the relaxation time τ of the heat flux is negligible or when the time variation
of the heat flux is slow this equation reduces to the usual Fourier law, which gives excellent
agreement with experiment for most practical problems.
Introduction of (2.4) into (2.2) results in a hyperbolic equation of the telegrapher type
∂ 2 T ∂T
τ 2 +
− χ∇ 2 T = 0.
(2.5)
∂t
∂t
By establishing (2.5) it is assumed that τ and χ are constant, where the quantity χ = λ/ρcv
designates the heat diffusivity. The dynamical properties of this equation have been thoroughly
analysed (e.g. Jou et al 1988a, Joseph and Preziosi 1990, 1991, Tzou 1997). However, the
thermodynamic consequences are less well known, and are therefore worth examining.
Consider an isolated undeformable wire of length L, directed along the x-axis, and take
an initial temperature profile of the form
2π nx
T (x, 0) = T0 + δT0 cos
(n = 0, 1, . . .)
(2.6)
L
where T0 is a uniform temperature reference. The problem is to study the decay of this
initial temperature profile towards equilibrium. We write the temperature perturbation δT as
δT = δT0 f (t) cos kx, with f (t) a function of time to be determined and k the wavenumber
k = 2π n/L; introducing this expression in (2.5) one obtains
f (t) = A exp(a+ t) + B exp(a− t)
where A and B are constants fixed by the initial conditions on T and ∂T /∂t, and
1
a± =
[−1 ± (1 − 4χk 2 τ )1/2 ].
2τ
For sufficiently high values of k such that 4χk 2 τ > 1 the decay is oscillatory. It is instructive
to evaluate the approach of entropy towards equilibrium in this case. According to classical
irreversible thermodynamics, the entropy S of the wire at temperature T is given by
Z L
T (x, t)
S(t) =
ρcv ln
dx
(2.7)
T0
0
Extended irreversible thermodynamics revisited (1988–98)
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Figure 1. The evolution of the classical equilibrium entropy SCI T during the equilibration of an
isolated system described by the Maxwell–Cattaneo equation is plotted as the discontinuous curve.
It is seen that SCI T has not a monotonically increasing behaviour, but decreases during some time
intervals. In contrast, the evolution of the extended entropy SEI T , defined in (2.39) and represented
by the continuous curve, increases monotonically.
where it has been assumed that the system is comprised between x = 0 and x = L and
ρcv is constant. The change of entropy in the course of time has been calculated by CriadoSancho and Llebot (1993a) by using (2.5) and is plotted in figure 1. It is seen that, instead
of being monotonically increasing, the classical entropy behaves in an oscillatory way (see
Jou et al (1993a) for an analysis of the discrete situation). Strictly speaking, this result is not
incompatible with the classical formulation of the second law, which states that the entropy
of the final equilibrium state must be higher than the entropy of the initial equilibrium state.
However, the non-monotonic behaviour of the entropy raises some questions about the localequilibrium formulation of the second law, which requires that the entropy production must be
everywhere positive at any time of the evolution. It is thus found that the Maxwell–Cattaneo
law (2.4), which has proved very successful for the description of many experiments on heat
waves, is not compatible with the local-equilibrium version of the second law. It is shown
below (see expression (2.39)) that one may obtain a monotonic behaviour for the generalized
entropy, compatible with the evolution stemming from the law (2.4) by assuming that the
entropy depends not only on the temperature but also on the heat flux.
The above example, and others involving shear viscous pressure (Fort and Llebot 1996),
clearly show that a description of fast phenomena requires the introduction in the expression of
the entropy of supplementary variables like heat flux, or, more generally, higher-order fluxes
or other related non-equilibrium variables.
2.2. Generalized entropy and entropy flux in a one-component fluid
As in CIT, EIT assigns a central role to entropy. It is assumed that the entropy of a onecomponent fluid depends locally not only on the classical conserved variables, such as internal
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energy u, specific volume v, etc but also on the fluxes, namely q (heat flux), pν (bulk viscous
0
pressure) and P ν (deviatoric part of the viscous pressure tensor). A short historical overview
of this hypothesis and of the development of the theory may be found in Jou et al (1988a).
For the sake of completeness, let us recall the basic balance laws of mass, momentum and
energy for a one-component fluid:
∂ρ
= −∇ · (ρ ν )
∂t
∂(ρ ν )
= −∇ · (P + ρ νν ) + ρ F
∂t
∂(ρu)
= −∇ · (q + ρuν ) − P T : ∇ ν
∂t
(2.8)
(2.9)
(2.10)
where ν is the barycentric velocity, F the external body force such as gravity or the Lorentz
force due to electromagnetic fields, and an upper index T denotes transposition. The total
pressure tensor P is related to the viscous pressure P ν by P = pU + P ν where p is the
equilibrium pressure and U the identity tensor; as usual, P ν is split into two parts: a bulk
pressure p ν (=
1
3
0
0
trace P ν ) and a deviatoric part P ν so that P ν = pν U + P ν .
0
The generalized entropy s(u, v, q , pν , P ν ) is supposed to possess the following properties:
(i) it is an additive quantity, (ii) it is a convex function of the whole set of variables, and (iii) its
rate of production is locally positive. The importance of these three properties cannot be
assessed a priori, but only from their consequences, which will be analysed later.
In differential form, the entropy can be written as
0
0
ds = θ −1 du + θ −1 π dv − θ −1 vα0 dpν − θ −1 v α1 · dq − θ −1 v α2 : dP ν
(2.11)
where all the extensive quantities s, u, v are measured per unit mass.
In analogy with the classical theory, we define an absolute temperature θ and a generalized
pressure π as partial derivatives of the entropy with respect to the internal energy and the
volume, respectively. These quantities contain non-equilibrium contributions, which will be
discussed in detail in section 3. In (2.11) we have assumed that
0
∂s/∂u = θ −1 (u, v, q , pν , P ν )
0
∂s/∂v = θ −1 π(u, v, q , pν , P ν )
0
(∂s/∂ q ) = −θ −1 v α1 (u, v, q , pν , P ν )
(2.12)
0
(∂s/∂p ν ) = −θ −1 vα0 (u, v, q , pν , P ν )
0
0
0
(∂s/∂ P ν ) = −θ −1 v α2 (u, v, q , pν , P ν ).
The minus sign and the factor θ −1 v in the three last relations are introduced for convenience.
0
In (2.12), α1 , α0 , and α2 are vector, scalar, and tensor fields, respectively. For simplicity we
assume for the moment that the equations of state (2.12) do not contain nonlinear terms in the
0
0
fluxes. It follows that α1 = α10 q , α0 = α00 pν , α2 = α21 P ν , where α10 , α00 , and α21 are scalar
functions of u and v, and at the same order of approximation, θ −1 = θ −1 (u, v) = T −1 (u, v) and
π = π(u, v) = p(u, v) where T and p are the local-equilibrium temperature and pressure,
respectively. The coefficients α00 , α10 , and α21 are identified in physical terms in the next
paragraph.
Extended irreversible thermodynamics revisited (1988–98)
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From (2.11) and the balance equations of energy and mass (2.8), (2.10), one obtains for
ṡ, the material time derivative of s,
0
0
ρ ṡ = −T −1 ∇ · q − T −1 pν ∇ · ν − T −1 P ν : V − T −1 α00 pν ṗν
0
0
−T −1 α10 q · q̇ − T −1 α21 P ν : (P ν ).
(2.13)
0
0
in which V is the deviatoric traceless part of the symmetric velocity gradient tensor and (P ν ).
0
denotes the material time derivative of P ν . This equation can be cast in the general form of a
balance equation
ρ ṡ + ∇ · J s = σ s
(2.14)
on the condition that we identify the expressions for the entropy flux J s and the entropy
production σ s .
A general expression of the entropy flux J s for isotropic systems is, up to second order
in the fluxes,
0
J s = θ −1 q + β 0 p ν q + β 00 P ν · q
0
(2.15)
00
where the coefficients β and β are functions of u and v. In the linear approximation in the
fluxes, the second and third terms in the right-hand side are negligible and θ can be identified
with the local-equilibrium temperature, so that (2.15) reduces to the expression of the classical
theory of irreversible processes, namely J s = T −1 q . The non-classical terms of J s have
been the subject of recent studies, both from a macroscopic point of view, by analysing their
connection with the non-local terms in the evolution equations of the fluxes, as well as from
the viewpoint of information and kinetic theories (Domı́nguez-Cascante and Jou 1995, Lebon
et al 1994b, Nettleton 1992, Nyiri 1991).
2.3. Entropy production and evolution equations for the fluxes
The entropy production is directly derived from the entropy balance equation (2.14) after the
expressions of ṡ and J s have been used. Introduction of (2.13) and (2.15) into (2.14) leads to
the following expression for the entropy production:
0
σ s = q · [∇T −1 + ∇ · (β 00 P ν ) + ∇(β 0 pν ) − T −1 α10 q̇ ]
+p ν [−T −1 ∇ · ν − T −1 α00 ṗν + β 0 ∇ · q ]
0
0
0
0
+P ν : [−T −1 V − T −1 α21 (P ν ). + β 00 (∇ q )s ].
(2.16)
This expression has the structure of a bilinear form
0
0
σ s = q · X1 + p ν X0 + P ν : X 2
(2.17)
0
consisting of a sum of products of the fluxes q , p ν , and P ν and their conjugate generalized
0
forces X1 , X0 , and X 2 . The latter follow from direct comparison of (2.17) with (2.16). They
are similar to the expressions obtained in CIT but contain additional terms depending on the
time and space derivatives of the fluxes.
0
Upon defining the proper form of the forces X1 , X 2 , and X0 , it can be noted that there
0
exists a class of transformations of the time derivatives of q and P ν which leave the entropy
production invariant. They allow us to introduce frame-indifferent time derivatives, for instance
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the Jaumann or corotational derivative, which are necessary in the analysis of rheological
equations and in the discussion of the invariance properties of the equations of the fluxes (Jou
et al 1988a, 1996a, Lebon and Boukary 1988).
To obtain evolution equations for the fluxes compatible with the positiveness of σ s , we
0
express the forces X1 , X0 , X 2 as linear functions of the fluxes. As a consequence, we write
X1 = µ1 q ,
0
X0 = µ0 pν ,
0
X2 = µ2 P ν
(2.18)
where the coefficients µi may depend on u and v but not on the fluxes.
When expressions (2.18) are introduced into (2.17), we are led to
0
0
σ s = µ1 q · q + µ0 pν pν + µ2 P ν : P ν .
(2.19)
The requirement that σ s must be positive yields the restrictions µ1 > 0, µ0 > 0, µ2 > 0.
The relations (2.18) can be generalized to the nonlinear domain, but this opens new
questions. To be more explicit, let us consider as a simple example the nonlinear law with
saturation behaviour
X1
q
q=
(2.20)
µ1 1 + aX12
with the corresponding entropy production given by
σ s = q · X1 =
X2
q 1
.
µ1 1 + aX12
(2.21)
The quantity σ s is clearly real and positive for any value of X1 provided that µ1 and a are
positive. However, if one expands the square root in series of X12 one has
X12
1 2 1 2 4
s
σ = q · X1 =
(2.22)
1 − aX1 + a X1 + · · ·
µ1
2
4
from which we notice that by truncating the expansion a finite number of terms it may occur
that σ s could become negative. Therefore, if one deals with a series expansion, the condition
σ s > 0 only imposes restrictions on the sign of the quadratic term, but not of the fourth and
higher-order terms in X1 (see also Garcı́a-Colı́n and Uribe (1991) and López de Haro et al
(1993)). Let us return to relation (2.16). Identifying in it the forces as the conjugate terms of
the fluxes and substituting these expressions in (2.18), one obtains the following set of linear
evolution equations:
0
∇T −1 − T −1 α10 q̇ = µ1 q − β 00 ∇ · P ν − β 0 ∇pν
−T −1 ∇ · ν − T −1 α00 ṗν = µ0 pν − β 0 ∇ · q
0
(2.23)
(2.24)
0
0
−T −1 V − T −1 α21 (P ν ). = µ2 P ν − β 00 (∇ 0 q )s
0
(2.25)
0
after the nonlinear contributions in (∇u) · P ν , (∇u) ·q , (∇ν) ·q , (∇ν) · P ν have been neglected.
The main features of this thermodynamic formalism can be summarized as follows: (i) the
positiveness of the coefficients µ1 , µ0 , and µ2 and (ii) the fact that the same coefficients β 0
and β 00 appear in the second-order terms of the entropy flux and the coefficients of the cross
terms in the evolution laws (2.23)–(2.25). These results are confirmed by the kinetic theory,
as shown in chapter 3 of Jou et al (1996a).
Extended irreversible thermodynamics revisited (1988–98)
1045
Several coefficients appear in (2.23)–(2.25) which can be identified on physical grounds.
Assume first a situation characterized by stationary and homogeneous fluxes (i.e. their time
and space derivatives are zero). Equations (2.23)–(2.25) then reduce to
∇T −1 = µ1 q ,
−T −1 ∇ · ν = µ0 pν ,
0
0
−T −1 V = µ2 P ν .
(2.26)
Comparison with the usual Fourier and Newton–Stokes laws,
q = −λ∇T ,
0
pν = −ζ ∇ · ν ,
0
P ν = −2ηV
2 −1
(2.27)
−1
−1
leads to the identifications µ1 = (λT ) , µ0 = (ζ T ) , and µ2 = (2ηT ) , with λ, ζ , and η
the thermal conductivity, bulk viscosity and shear viscosity, respectively.
In the next step, if we assume non-stationary (but homogeneous) fluxes, equations (2.23)–
(2.25) simplify as
∇T −1 − T −1 α10 q̇ = (λT 2 )−1 q
−T −1 ∇ · ν − T −1 α00 ṗν = (ζ T )−1 pν
0
(2.28)
(2.29)
0
0
−T −1 V − T −1 α21 (P ν ). = (2ηT )−1 P ν .
(2.30)
These equations can be identified with the so-called Maxwell–Cattaneo laws expressed by
τ1 q̇ + q = −λ∇T
τ0 ṗ ν + p ν = −ζ ∇ · ν
0
0
(2.31)
(2.32)
0
τ2 (P ν ). + P ν = −2ηV
(2.33)
where τ1 , τ0 , and τ2 are the relaxation times of the respective fluxes. We are then led to the
identifications α10 = τ1 (λT )−1 , α00 = τ0 ζ −1 , and α21 = τ2 (2η)−1 . In terms of the transport
coefficients λ, ζ , η, and the relaxation times τ1 , τ0 , and τ2 , the evolution equations (2.23)–(2.25)
take the form
0
τ1 q̇ = −(q + λ∇T ) + β 00 λT 2 ∇ · P ν + β 0 λT 2 ∇pν
τ0 ṗ ν = −(pν + ζ ∇ · ν ) + β 0 ζ T ∇ · q
0
0
(2.34)
(2.35)
0
τ2 (P ν ). = −(P ν + 2ηV ) + 2β 00 ηT (∇ 0 q )s .
(2.36)
Of course, equations (2.34)–(2.36) must be supplemented with suitable initial and boundary
0
conditions. Some consequences of the crossed terms coupling q and P ν have recently been
studied by Pérez-Guerrero (1997) in the context of thermal wave propagation in systems under
shear, where the velocity of waves is different in the shear and normal directions.
Within the linear approximation, the generalized Gibbs equation (2.11) has the form
0
τ1
τ0 ν ν
τ2 0 ν
p dp −
q · dq −
P : dP ν .
(2.37)
2
ρλT
ρζ T
2ρηT
To show that this result circumvents the problem raised in section 2.1, let us still reconsider
heat transport in a rigid body when the evolution of q is governed by the Maxwell–Cattaneo
equation. But now the entropy is no longer given by its local-equilibrium value but is obtained
from expression (2.37), which for pure heat conduction reduces to
τ1
q · dq .
(2.38)
ds = T −1 du −
ρλT 2
After integration, (2.38) can be cast in the form
τ1
q·q
(2.39)
ρs(u, q ) = ρseq (u) −
2λT 2
ds = T −1 du + pT −1 dv −
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D Jou et al
where ρseq (u) is the local-equilibrium entropy depending only on u. The corresponding
entropy production is
σs =
1
q · q.
λT 2
(2.40)
In figure 1 the extended entropy (2.39) and the local-equilibrium entropy are plotted versus
time during the decay of the temperature perturbation towards equilibrium (Criado-Sancho
and Llebot 1993a). It is seen that the local-equilibrium entropy behaves in an oscillatory way,
whereas the extended entropy (2.39) increases monotonically. It can thus be concluded that
the Maxwell–Cattaneo equation (2.4) for the heat flux, which is not compatible with the localequilibrium thermodynamic theory, as the condition σ s > 0 is not always fulfilled, satisfies
fully the positiveness of σ s in the extended thermodynamic formalism.
Up to now, we have presented EIT as a generalization of the classical theory of irreversible
processes. A similar view, known as the wave approach to thermodynamics, was proposed by
Gyarmati (1977); discussions about its relation with EIT may be found in Garcı́a-Colı́n and
Rodrı́guez (1988) and Markus and Gambar (1989), and some applications were performed
by Markus (1989) and Verhás (1989). Other techniques, such as those followed in rational
thermodynamics, can be used to formulate EIT, as shown in the next section.
2.4. Rational extended thermodynamics
Here, we propose a version of EIT formulated along the line of thought of RT, which we call
rational extended thermodynamics (RET). The formalism of RT was essentially developed by
Coleman, Truesdell and Noll in the 1960s (see Truesdell (1984) for an outstanding monograph)
and follows a rationale drastically different from CIT. Its main objective is to provide a method
for deriving constitutive equations. Among the basic hypotheses underlying RT we can choose
from are that: (i) absolute temperature and entropy are considered primitive concepts; (ii) it is
assumed that systems have a memory, i.e. their behaviour at a given instant of time is determined
not only by the values of the characteristic parameters at the present time, but also by their past
history and (iii) the mathematical formulation of the second law of thermodynamics, which
serves essentially as a restriction on the form of the constitutive equations, is expressed by
means of the Clausius–Duhem inequality. Although we borrow some methods and concepts
from rational thermodynamics, we depart radically from it. Essentially, the choice of the
independent variables is different: in RT the variables are the histories of the classical state
variables u, v, . . . ; moreover, the response of the material system is described by constitutive
functionals rather than by evolution differential equations. Here rational thermodynamics is
utilized as a method and not as a theory.
To illustrate our approach, we first consider the problem of a viscous heat-conducting
fluid in motion. The space of the variables, denoted V , is formed by the union of the space
of the classical variables C (the density ρ(= v −1 ), the specific internal energy u, the velocity
ν ) and the space of the fluxes F (here the heat flux q , the bulk viscous pressure p ν , and the
0
shear viscous pressure P ν ). The evolution of the classical variables is governed by the balance
equations of mass, momentum and energy. Denoting these variables by 8, one may cast these
balance laws in the general form
ρ 8̇ = −∇ · J 8 + σ 8
(2.41)
where J 8 and σ 8 stand for the corresponding flux and source. Unlike RT, the source terms in
the momentum and energy equations are known functions of space and time.
Extended irreversible thermodynamics revisited (1988–98)
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0
Concerning the extra variables q , pν and P ν , it is natural to suppose that they obey
evolution equations similar to (2.41), namely
ρ q̇ = −∇ · Q + σ q
ρ ṗν = −∇ · j ν + σ ν
0
0
(2.42)
0ν
ρ(P ) = −∇ · J + σ .
ν .
ν
Q is a tensor of rank two representing the flux of the heat flux, and σ q is a vector corresponding
to the supply of heat flux, j ν is a vector denoting the flux of the scalar viscous pressure and
0
σ ν the corresponding scalar source term, J ν is a third-rank tensor designating the flux of the
0
traceless viscous pressure tensor, and σ ν is its source term. At this stage of the analysis these
quantities are not determined and must be specified by means of constitutive relations.
The evolution equations (2.42) and the constitutive relations are not arbitrary. They have
to comply with the constraints of Euclidean invariance (criterion of objectivity), positiveness
of the rate of entropy production, and convexity of entropy. As a consequence of objectivity,
the material time rates must be replaced by objective ones as recalled in section 2.
The five balance laws of mass, momentum and energy and the nine evolution
equations (2.42) form a set of 14 relations for the 14 unknowns: ρ (scalar), u (scalar), ν
0
(vector), q (vector), p ν (scalar), P ν (traceless symmetric tensor) when we express the fluxes
0
0
Q, j ν , J ν and the sources σ q , σ ν , σ ν in terms of the whole set of variables V . This implies
that s and J s are given by constitutive relations of the form s = s(V ), J s = J s (V ).
To satisfy the second law of thermodynamics, it is assumed that there exists a regular and
continuous function s, called entropy, which obeys a balance law given by
ρ ṡ + ∇ · J s = σ s > 0
(2.43)
where J s is the entropy flux and σ s the non-negative rate of entropy production. As in rational
thermodynamics, the non-negative property of σ s is used to place restrictions on the constitutive
equations. The rate of entropy production is calculated by performing the operations indicated
on the left-hand side of (2.43).
At this point, let us emphasize the main differences between EIT and rational
0
thermodynamics. First, whereas in the latter theory the quantities q , pν , and P ν are given
by constitutive relations, in EIT they are counted among the set of independent variables.
Secondly, the balance laws of momentum and energy are not regarded as mere definitions of
the body force and energy supply; in EIT these quantities are given a priori. Thirdly, the
second law is not in the form of the Clausius–Duhem inequality as the entropy flux is not
imposed a priori to be given by the ratio of the heat flux and the temperature, but may contain
extra terms. Fourthly, the entropy is assumed to depend on the fluxes.
Here, for the sake of simplicity, we restrict our attention to the problem of heat conduction
in a rigid isotropic solid at rest. The relevant variables are now the internal energy u and the heat
flux q . To take into account the restrictions placed by the second law (2.43), we follow the well
known method of Lagrange multipliers proposed by Liu (1972) and widely used by Ruggeri
(1989) and Müller and Ruggeri (1993) in their formulation of extended thermodynamics.
According to this technique, we include the constraints introduced by the energy balance
and by the evolution equation of q via the Lagrange multipliers 30 (u, q ) and 31 (u, q ), so that
inequality (2.43) will take the form
ρ ṡ + ∇ · J s − 30 (ρ u̇ + ∇ · q ) − 31 · (ρ q̇ + ∇ · Q − σ q ) > 0.
(2.44)
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D Jou et al
s and J s are, at this stage of the analysis, unknown functions of u and q . It should be stressed
that the present approach differs from that of sections 2.2 and 2.3, as the balance laws are now
considered as constraints for inequality (2.43) to hold. By differentiating s and J s with respect
to u and q , and rearranging the various terms one obtains from (2.44)
∂s
∂Js
∂Js
∂s
− 30 ρ u̇ +
− 31 · ρ q̇ +
· ∇u +
: ∇q
∂u
∂q
∂u
∂q
−30 ∇ · q − 31 · (∇ · Q) + 31 · σ q > 0.
(2.45)
Since u̇ and q̇ can be given arbitrary and independent values, the positiveness of (2.45) requires
that
∂s
∂s
1
= 30 = ;
= Λ1
(2.46)
∂u
θ
∂q
where we have identified 30 with θ −1 as it has the dimension of the inverse of a temperature.
Assuming that the material is isotropic and omitting non-local and nonlinear effects, ∂ J s /∂ q
must be proportional to the unit tensor U . Under the assumption that Q does not contain
nonlinear terms in qq , the only terms containing ∇ q in (2.45) are
s
∂J
− 30 U : ∇ q .
(2.47)
∂q
Since ∇ q is arbitrary, the positiveness of (2.45) implies
1
J s = 30 q = q .
(2.48)
θ
This is as far as we may go without further hypotheses on the form of Q and σ q . Omitting
second- and higher-order terms in the flux, the most general expressions of Q and σ q are
simply
Q = a(u)U ;
σ q = −b(u)q
(2.49)
with a(u) and b(u) undetermined functions of u. Introducing (2.47)–(2.49) into (2.45) it is
found that
s
∂J
∂a
− 31
· ∇u − b31 · q > 0.
(2.50)
∂u
∂u
Since according to (2.48) ∂ J s /∂u = (∂θ −1 /∂u)q , and (2.50) is linear in ∇u, the
positiveness of (2.50) requires that the coefficient of ∇u is zero; as a consequence,
q
.
(2.51)
31 =
(∂a/∂θ −1 )
To recover the Maxwell–Cattaneo equation, let us define τ1 and λ by τ1 = ρ/b and
λ = −τ1 /(ρθ 2 )(∂a/∂θ −1 ). Substituting these expressions in the first of (2.42) after using
(2.49), one finds that indeed
τ1 q̇ = −λ∇θ − q .
(2.52)
−1
It remains to identify the Lagrange multiplier 31 ; from the expression for ∂a/∂θ , it is directly
seen that
τ1
31 = −
q
(2.53)
ρλθ 2
which is the result for ∂s/∂ q given in (2.38). Finally, by substituting the expression of b
and (2.53) in (2.50), it is found that (λθ 2 )−1 q · q > 0, from which it results that λ > 0.
According to the first equality in (2.46) and (2.53), the Gibbs equation may be written as
τ1
ds = 30 du + 31 · dq = θ −1 du −
q · dq
(2.54)
ρλθ 2
Extended irreversible thermodynamics revisited (1988–98)
1049
which is equivalent to (2.38).
The above procedure can be easily generalized by including other dissipative fluxes, like
the viscous pressure tensor and the bulk viscous pressure. Recall that the above analysis rests
on very simple hypotheses reflected by (2.49). There is no difficulty in generalizing them by
taking, for instance, for Q the relation Q = a(u)U + a2 (u)qq + a3 (u)∇ q , including nonlinear
and non-local contributions. However, we will not deal here further with this specialized topic
(Lebon et al 1998).
It is interesting to summarize the main differences between the standard and rational
presentations of EIT in sections 2.3 and 2.4: (i) in RET, the restrictions on the balance equations
are explicitly taken into account in the formulation of the second law by means of Lagrange
multipliers; (ii) the Gibbs equation is not postulated a priori, as in the previous formulation,
but it is derived from the restrictions of the second law. However, at the linear approximation,
the results of both approaches are identical both for the evolution equations of the fluxes and for
the Gibbs equation, since the Lagrange multipliers can be identified with the derivatives of the
entropy with respect to the basic variables. In the nonlinear range, the complete equivalence
of both approaches is not yet established. In our opinion, the existence of both approaches
is useful and stimulating, as each point of view has its own advantage. For instance, the use
of Lagrange multipliers in RET is elegant and appealing, but the standard description of EIT
provides for them a physical identification in a more direct way. Another example is found in
the analysis of fluctuations, which is more straightforward in the standard version of EIT.
Let us mention that an analogous analysis to the one performed here for continuous
systems may be carried out for discrete systems (Muschik 1993, Muschik and Kaufmann
1994, Muschik and Domı́nguez-Cascante 1996). These latter authors use as independent
variables the incoming heat, and the outgoing heat, in such a way that one has more than one
contact temperature and one contact pressure.
In the version of EIT proposed by Müller and Ruggeri (1993), and Wilmanski (1998)
one takes advantage of mathematical theorems related to symmetric hyperbolic differential
equations which guarantee the properties of existence, uniqueness and well-posedness of their
solutions. This is an interesting perspective, because it allows one to show the existence of a
non-equilibrium entropy by starting from a set of evolution equations, and therefore it outlines
the deep connection between dynamics and thermodynamics. The reader is referred to Müller
and Ruggeri (1993), Strumia (1988) and Levermore (1996) for more details.
3. Non-equilibrium equations of state
In section 2, we postulated the existence of a generalized entropy compatible with some
classes of evolution equations for the fluxes. Otherwise stated, our formalism aims to describe
processes which are compatible with the existence of a non-equilibrium entropy whose rate of
production is non-negative. Once the expression of the entropy is known, there is no difficulty in
deriving the corresponding equations of state which are directly obtained as the first derivatives
of the entropy with respect to the basic variables. A natural question concerns the physical
meaning of these equations of state which, of course, depend on the fluxes and therefore
differ from their analogous local-equilibrium expressions. In classical thermodynamics, it is
known that the derivative of the entropy with respect to the internal energy (by keeping fixed
the volume and the composition of the system) is the absolute temperature; the derivatives
with respect to the volume and to the number of moles yield the equilibrium pressure and
the chemical potentials (divided by the absolute temperature), respectively. It may then be
asked whether the derivatives of the generalized entropy introduced in EIT still allow one to
define an absolute non-equilibrium temperature, as well as a non-equilibrium pressure and
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D Jou et al
Figure 2. The local-equilibrium entropy Seq (u) (upper curve) and the generalized non-equilibrium
entropy s(u, q ) (lower curve) are shown. The non-equilibrium θ and equilibrium T temperatures
are given by the inverse of the slopes of the curves at points A and B, respectively.
non-equilibrium chemical potentials. This is a very subtle and not completely solved problem
which has, however, received partial answer in recent years, in that some specific gedanken
experiments reflecting, in particular, the differences between the generalized temperature and
the local-equilibrium temperature were proposed (Jou and Casas-Vázquez 1992, 1993, CasasVázquez and Jou, 1989, 1994) and a real experiment was interpreted (Luzzi et al 1997a). Since
this is a fundamental question, it deserves detailed attention. With this objective in mind, let
us try to better apprehend the physical meaning of the generalized entropy defined by (2.37).
3.1. Physical interpretation of the non-equilibrium entropy
Consider a volume element of a homogeneous fluid which is sufficiently small so that within it
the spatial variations of pressure and temperature are negligible; if the fluid element is subject
0
to a heat flux q and a viscous pressure P ν (viscous bulk effects are ignored), it is then asked
which entropy may be ascribed to it. To answer this question, the volume is suddenly isolated,
i.e. bounded by adiabatic and rigid walls, and allowed to decay to equilibrium. The decay of q
0
and P ν to their final vanishing equilibrium values is accompanied by a production of entropy,
so that the final equilibrium entropy value is given by
Z ∞
s
ρeq,f = ρsi +
σ s dt.
(3.1)
0
Indices i and f refer to the initial non-equilibrium state and the final equilibrium state
respectively, s is the entropy per unit mass and σ s is the rate of entropy production per unit
volume.
When the flux variables are fixed, the entropy can be represented by a surface in the space
s,u,v; equilibrium corresponds to the vanishing of the fluxes while a non-equilibrium surface
is characterized by nonzero values of the fluxes. In figure 2, the non-equilibrium state (A) and
the local-equilibrium state (B) corresponding to the same values of u and v are represented.
The second term in the right-hand side of (3.1) corresponds to the non-compensated heat
introduced by Clausius and will be commented on below. Its explicit form in the case of the
Maxwell–Cattaneo equations may be obtained in a rather straightforward way. Let t = 0 be
the instant at which the volume element is isolated. According to (2.19) and the identifications
Extended irreversible thermodynamics revisited (1988–98)
1051
µ1 = (λT 2 )−1 and µ2 = (2ηT )−1 , σ s is given by
0
0
σ s = (λT 2 )−1 q · q + (2ηT )−1 P ν : P ν .
(3.2)
0
If the decay of q and P ν is governed by the Maxwell–Cattaneo equations (2.31)–(2.33), we
have
q (t) = q (0) exp(−t/τ1 ),
0
0
P ν (t) = P ν (0) exp(−t/τ2 ).
(3.3)
Inserting these expressions into (3.2) and integrating with respect to the time, one obtains for
the non-equilibrium entropy s in the steady state
0
0
ρs = ρseq − (τ1 /2λT 2 )q (0) · q (0) − (τ2 /4ηT )P ν (0) : P ν (0)
(3.4)
which is the integrated form of the entropy (2.37) in the absence of bulk viscous pressure. The
above derivation is suggestive as it assigns a meaning to the non-classical terms in the equation
for the entropy, by relating them to a physical operational definition.
However, it must be realized that the above procedure exhibits three limitations: it is
0
based on the hypotheses that q and P ν are the relevant additional variables; it is assumed that
0
q and P ν decay exponentially; and it is restricted to a given non-equilibrium process (namely,
relaxation of the fluxes after sudden isolation of the elementary volume). It would therefore
be of interest to get rid of these restrictions.
In that respect, some ideas have been proposed by Eu (1992, 1995a, b) who started from
the well known Clausius inequality written for a cycle process:
I
dQ
6 0.
(3.5)
T
T is the absolute temperature of the heat reservoir which is in thermal contact with the system
during the infinitesimal process of exchange of a quantity of heat dQ. Inequality (3.5) may
still be expressed as
I
dQ
−N ≡
60
(3.6)
T
whereHN is the (positive) uncompensated heat. If N is expressed in the form of a cyclic integral
N = dN, then the Clausius integral may be rewritten as
I dQ
+ dN = 0.
(3.7)
T
The vanishing of integral (3.7) for arbitrary cycles implies the existence of the exact differential
of a given quantity which will be called the generalized entropy in our formalism (or the
calortropy in Eu’s work); the latter is defined as
dQ
dS =
+ dN.
(3.8)
T
Note that neither dQ/T nor dN are exact differentials, but only their sum is so. Furthermore,
dN is always positive, and it vanishes for reversible processes. In the latter case, dN = 0,
dQ = dQrev , and S = Seq , so that one recovers the usual definition
dQrev
.
(3.9)
T
The present argument is more general than the one given before, as it does not require us
to specify a priori either the nature of the non-equilibrium variables, or the dynamics of the
dSeq =
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D Jou et al
variables, or the nature of the process being involved. As an illustration, assume that the extra
variables are q and P ν . Recalling that dQ is given by the first law, one has
dU + pdV
dQ
=
(3.10a)
T
T
with V the total volume of the system. Comparison of (3.8) with the Gibbs equation (2.37)
suggests that
0
0
(3.10b)
dN = −T −1 V α1 · dq − T −1 V α 2 : dP ν .
Indeed, as we commented above, the extra terms in the non-equilibrium entropy may be related
to the uncompensated heat generated during the relaxation to equilibrium; it is logical that they
are related to dN. It should be stressed that Eu’s presentation is general and rather abstract
but does not help in selecting the variables. The choice of the relevant variables must always
be motivated by experimental and (or) microscopic considerations, as in EIT wherein the
choice of q and P ν is determined from the dynamical equations (2.34)–(2.36) which have
found experimental confirmation. Another difficulty with respect to Eu’s argument is that it
supposes the existence of an irreversible cycle. This is clearly not guaranteed as it implies the
selection of a complete set of relevant variables. If the set is not complete, a cycle in the space
of these variables would not correspond to a true cycle of the system, i.e. the final state could
be different from the initial state despite the fact that the set of chosen variables take the same
values in the initial and final states.
Note that the family of states whose adiabatic projection (i.e. the set of states which, when
isolated, decay to a given equilibrium state) is a given equilibrium state may be considered
as a fibre in the general thermodynamic space, which by this procedure is given a structure
of a fibred space (Grmela 1993d, Banach and Pierarski 1996, Chen 1997). The adiabatic
projection is not the only conceivable projection (Muschik 1990a, b): it is also possible
to think about an isothermal projection or other projections, which may be of interest, for
instance, in microscopic analyses of EIT based on projection-operator techniques (Ichiyanagi
1990a, b). In these cases it is important to specify carefully which is the equilibrium state
taken as a reference, especially in nonlinear situations, where different reference states may
lead to different results for the evolution equations and the equations of state.
3.2. Non-equilibrium equations of state for temperature, pressure and chemical potential
Up to now, we have neglected the second-order contributions of the fluxes in the equations of
state (2.12) for temperature and pressure. Having identified in section 2.3 the parameters α10 ,
α00 , and α21 in physical terms, we are now in a position to evaluate such contributions. Up to
second-order terms in the fluxes, the Gibbs equation (2.11) may be written as
0
vτ1
vτ0 ν ν
vτ2 0 ν
ds = θ −1 du + θ −1 π dv −
(3.11)
p dp −
q · dq −
P : dP ν
2
λT
ζT
2ηT
where, in contrast to (2.37), we have re-introduced in the first two terms the generalized absolute
temperature θ and the thermodynamic pressure π instead of their respective local-equilibrium
approximations T and p. From the integrability condition of (3.11) and keeping in mind that
for vanishing fluxes one must recover the local-equilibrium values of T and p, one obtains
1 ∂(vτ1 /λT 2 )
∂(vτ0 /ζ T ) ν 2 ∂(vτ2 /2ηT ) 0 ν 0 ν
−1
−1
q·q+
P :P
(p ) +
θ =T −
2
∂u
∂u
∂u
+O(3)
(3.12)
1 ∂(vτ1 )/λT 2 )
∂(vτ0 /ζ T ) ν 2 ∂(vτ2 /2ηT ) 0 ν 0 ν
q·q+
(p ) +
P :P
θ −1 π = T −1 p −
2
∂v
∂v
∂v
+O(3).
Extended irreversible thermodynamics revisited (1988–98)
1053
0
Had we included second-order flux terms in α0 , α1 , α2 , then (3.12) would contain third-order
contributions which are here supposed negligible. These expressions can be viewed as nonequilibrium equations of state for the temperature and pressure.
Let us now discuss in more detail their physical content and start with the first of
equations (3.12), related to the temperature θ . According to (3.11), the latter is defined by
∂s
−1
.
(3.13)
θ =
∂u v,q,pν ,P0 ν
If the entropy is known in terms of the various variables, (3.13) gives directly the equation of
state of θ. Conversely, when all the equations of state are known, they may be integrated to
obtain the expression of the entropy.
From now on, we focus on situations where the heat flux q is the only non-equilibrium
flux. The first of expressions (3.12) reduces to
1 ∂(τ1 v/λT 2 )
q·q
(3.14)
2
∂u
wherein T (u) depends only on the internal energy and not on the heat flux. To illustrate the
difference between θ and the local-equilibrium temperature T , project the surface s(u, q ) onto
the plane (s, u) (see figure 2). It follows from (3.13) that the tangent to the curve s(u) at B is
related to the local-equilibrium T while the tangent at A is a measurement of the generalized
temperature θ.
For monatomic ideal gases obeying the Boltzmann equation, a well known result of kinetic
theory is that τ1 /λT 2 = (2/5)(m/kB2 T 3 n), with m the mass of a molecule, kB the Boltzmann
constant and n the number density, so that the inverse of the temperature is given by
2 ρ
q·q
(3.15)
θ −1 (u, q ) = T −1 (u) +
5 p3 T
after making use of p = nkB T . This expression provides an explicit estimation of the difference
between T and θ.
For metallic rigid conductors, (3.14) yields (Jou and Casas-Vázquez 1992)
θ −1 (u, q ) = T −1 (u) −
θ −1 (u, q ) = T −1 (u) +
9 m2 εF
q·q
π 4 nkB4 T 5
(3.16)
with m the electron mass, n the electron number density, and εF the Fermi energy of the metal.
Since the coefficients of the term in q · q are very small, it turns out that in many practical
situations the corrective term may be neglected. Indeed, for CO2 at 300 K and 0.1 atm, and for
a heat flux of the order of 105 W cm−2 , T −θ = 9.6×10−2 K. However, in other circumstances
the difference is not minute: in nuclear collisions, as studied in section 16, T − θ is of the
order of 7% of T , and in radiation near the surface of stars, the difference is close to 3% (Fort
1997).
Another consequence of the non-equilibrium temperature is that even in steady situations
the relation between heat flux and temperature gradient should be written as q = −λ∇θ instead
of q = −λ∇T . This result can be used as the starting point of a better understanding of θ .
Consider the situation depicted in figure 3 (Jou and Casas-Vázquez 1992, Jou et al 1993). A
reference system 6r and the system under consideration 6s are thermally connected through
a highly conducting rod of length L laterally adiabatically insulated. We denote by qr , qs and
qrs the heat fluxes flowing through 6r , 6s and the rod, respectively. Then we introduce the
following statement: we say that 6r and 6s are in mutual thermal equilibrium under steady
conditions when qrs = 0. This does not necessarily imply that qr = 0 and qs = 0, i.e. the
two systems may be out of equilibrium notwithstanding the condition qrs = 0. Of course,
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D Jou et al
Figure 3. Illustration of the gedankenexperiment used to discuss the concept of non-equilibrium
thermodynamic temperature θ.
in complete equilibrium qr = qs = qrs = 0. If q = −λ∇θ , the heat exchange between 6r
and 6s is directly governed by the non-equilibrium temperature θ . If 6r and 6s are both at
equilibrium, θ coincides with the local-equilibrium temperature T . However, if 6r and 6s are
out of equilibrium, with qrs = 0, then one will observe that θr = θs rather than Tr = Ts .
To be more specific, assume that the system on the left is at equilibrium at temperature
T (figure 3). The system on the right is in a non-equilibrium steady state under a heat flux
qy generated by a temperature difference T 00 − T 0 . Both ends of the conducting rod are at
the same local-equilibrium absolute temperature T . According to the classical theory, no heat
flow will be observed from one system to another. In contrast, EIT predicts a flow qx from
the left to the right, which is proportional to the gradient of the non-equilibrium temperature
θ , namely (θr − θs )/L. Since the system 6r on the left is in equilibrium, one has θr = Tr ,
but θs = Ts [1 − γ qy2 ], with γ = (2/5)(ρ/p3 ) for ideal monatomic gases, so that θs < Ts .
Consequently, a heat flux qx will flow from the left to the right. This example corroborates
the statement that a thermometer will measure the generalized temperature rather than the
local-equilibrium one: heat transfer between the two systems will take place until they have
reached the same generalized temperature, rather than the same local-equilibrium temperature.
A confirmation can be found in the kinetic theory; indeed, it was shown (Casas-Vázquez and
Jou 1994) that when both systems 6s and 6r of figure 3 are composed of an ideal gas with
the same mean kinetic energy at the position of the thermal contact, energy flows from the
equilibrium gas to the gas submitted to the vertical heat flux.
An analogous phenomenon should appear when the non-equilibrium situation is produced
by other fluxes (as a shear viscous pressure or an electrical current) instead of a heat flux in
the system on the right of figure 3. Indeed, recent computer simulations have shown that in
the presence of a shear viscous pressure, heat flows between regions which, nevertheless, are
at the same local-equilibrium temperature (Todd and Evans 1997). Furthermore, there is some
experimental evidence from spectroscopic analyses in photoexcited plasma in semiconductors
submitted to an external electric field, that indeed the temperature appearing in the nonequilibrium distribution function should depend not only on the local-equilibrium variables
but also on the electrical current and heat flux (Luzzi et al 1997a).
The presence of a generalized temperature is not exclusive to extended irreversible
thermodynamics. In his entropy-free formulation of non-equilibrium thermodynamics,
Meixner (1973) postulated the existence of a dynamical temperature depending on the
interactions of the system with the outside. Müller (1971) introduced a ‘coldness’ function
assumed to depend on the empirical temperature and its time derivative: in a steady state,
Extended irreversible thermodynamics revisited (1988–98)
1055
the ‘coldness’ reduces to the local-equilibrium temperature, in contrast with the generalized
temperature appearing in EIT. More recently, Muschik (1980, 1990a, b) introduced the notion
of contact temperatures and explored the conceptual difficulties of their measurement; this
contact temperature is not identical to the local-equilibrium temperature. Finally, Keizer
(1987) proposed a non-equilibrium temperature defined as the derivative with respect to the
internal energy of a generalized entropy derived from statistical considerations about molecular
fluctuations; this temperature depends not only on the classical variables but also on the second
moments of fluctuations. It is worth stressing that the EIT generalized temperature may also be
expressed in terms of the second moments of the energy fluctuations, in analogy with Keizer’s
approach (see section 4.2).
The problems raised by the definition and measurement of a non-equilibrium temperature
have been contested in recent papers (Henjes 1992, Hoover et al 1992, Nettleton 1994, Eu
and Garcı́a-Colı́n 1996, Bhalekar and Garcı́a-Colı́n 1998). Apart from notation (one could use
T and Teq instead of θ and T ), a question under discussion is the nature of the variables to
be kept constant during the differentiation of the entropy in (3.13). With v and q fixed, one
recovers (3.12a). If, instead, one keeps constant v and the quantity (τ1 /ρλT 2 )1/2 q (Banach
and Pierarski 1996, Garcı́a-Colı́n and Micenmacher 1996), the derivatives of the generalized
entropy and of the local-equilibrium entropy coincide. Indeed, since
τv
q·q
(3.17)
s(u, v, q ) = seq (u, v) −
2λT 2
then, of course,
∂seq
∂s
=
.
(3.18)
∂u v,√τ/(λT 2 )q
∂u v
Still another possibility (Brey and Santos 1992) is to maintain fixed the temperature gradient
and, since (3.17) implies that for steady situations
τ vλ
∇T · ∇T
(3.19)
s = seq −
2T 2
one is led to (Casas-Vázquez and Jou 1994)
∂s
= T −1 + (1 + 2b)(τ λ/2ρcv T 3 )∇T · ∇T
(3.20)
∂u v,∇T
where b is the exponent which characterizes the dependence of τ with T according to τ ≈ T −b
(i.e. b = 0 for Maxwell molecules and b = 21 for hard spheres). In fact, it can be argued that
not all these definitions may be valid: recall, indeed, that in equilibrium thermodynamics
(∂s/∂u)v = T −1 , but if one keeps constant the pressure instead of the volume during the
differentiation, (∂s/∂u)p is different from T −1 ; clearly (∂s/∂u)p = T −1 [1 − (pvα/cp )]−1 ,
with α the coefficient of thermal expansion and cp the specific heat at constant pressure. At the
present time, it is not clear which among the above restrictions is the most suitable to define
appropriately the non-equilibrium temperature.
To illustrate the difference between the non-equilibrium temperature θ and the localequilibrium temperature T let us go back to figure 2, where the curves s(u, q ) are drawn for
two values of q . The curve for q = 0 corresponds to equilibrium states. The actual state of the
system is described by point A which is characterized by given values of u and q . According to
definition (2.12), the inverse of the non-equilibrium temperature θ −1 is given by the slope of the
curve s(u, q ) at point A. The accompanying local-equilibrium state obtained by isolating the
system and letting it decay to equilibrium is represented by the point B, located on the curves
seq (u), corresponding to q = 0. The inverse of the local-equilibrium absolute temperature T −1
is the slope of the curve seq (u) at point B. Note that θ −1 > T −1 . This is a rather general feature
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D Jou et al
because s(u, q ) 6 seq (u); it is also worth noticing that s(u, q ) → seq (u) when u → ∞ at
constant q (this results from the property that the value of τ v q 2 /λT 2 decreases with increasing
u at constant q ). Therefore, the curve s(u, q ) will be steeper than seq (u) yielding, generally,
θ 6 T . A similar inequality is found in computer simulations for fluids under a velocity
gradient (Baranyai and Evans 1991).
The definition of the non-equilibrium pressure requires subtler considerations. In fact,
0
rather than the fluxes themselves, one must use v q , vp ν , v P ν as independent variables. Indeed,
these variables are extensive in the following sense: if we have two systems of volumes V1 and
V2 crossed by the same heat flux q , the variable V q is additive, i.e. Vtot q = V1 q + V2 q although
q itself is not additive. The consequence of this choice will be reflected in the definition of
the non-equilibrium pressure, which is defined as the derivative of the entropy with respect to
the volume at constant v q rather than at constant q . Since the temperature is related to the
differential of the entropy at constant v, the fact that v q or q are kept constant does not modify
the results, because v is a constant.
Indeed, it must be recalled from equilibrium thermodynamics that the entropy is a
characteristic function (i.e. all conceivable thermodynamic information about the system is
ascertainable from it) on condition that it is expressed in terms of its extensive natural variables.
Therefore, the most suitable defintion for the non-equilibrium pressure is
∂s
−1
.
(3.21)
πθ =
∂v u,vq
Apparently, a conceptual problem arises, because in the classical theory the equilibrium
pressure is defined as one third of the trace of the equilibrium pressure tensor. Since p = 23 ρu
for an ideal gas, it follows that if one keeps ρu constant then the local-equilibrium pressure
cannot be dependent on q . In fact, it is easy to avoid this apparent contradiction if the pressure
tensor in the presence of a non-vanishing heat flux is assumed to have the form (Domı́nguez
and Jou 1995)
P = π U + α qq
(3.22)
where α is a phenomenological coefficient which may depend on u and v. Since, for an ideal
gas, the pressure tensor must satisfy tr P = 3p, we require that the coefficient α in (3.22) is
given by the condition tr P = 3π + αq 2 = 3p. The form (3.22) makes it clear that the fact that
the derivative of the entropy with respect to the volume is affected by the presence of q is no
contradiction of the fact that the trace of the tensor is not modified by the heat flux. Let us add
that the expression (3.22) is supported by microscopic analyses of electromagnetic radiation
(Domı́nguez-Cascante and Faraudo 1996) as it will be commented on in connection with the
Eddington factor in (6.43), and the Hamiltonian methods introduced in section 5.
It is helpful to write explicitly P in a simple situation, namely, when the heat flux has the
y direction. In this case one has
!
!
0
0
0
π 0 0
2
(3.23)
P = 0 π 0 + 0 αqy 0 .
0
0
0
0 0 π
This implies that the work of compression or of expansion will depend on the relative direction
between the axis of compression and the heat flux. This has been noted in some computer
simulations (Evans 1989, Baranyai and Evans 1991; see, however, Nettleton 1996a for an
alternative point of view), where the thermodynamic pressure is seen to agree with the minimum
eigenvalue of the pressure tensor. This is indeed the situation found in (3.23), because π < p
and therefore αq 2 > 0. A gedankenexperiment analogous to that in figure 3 could be devised
Extended irreversible thermodynamics revisited (1988–98)
1057
for the non-equilibrium pressure, by replacing the rigid conducting rod connecting both systems
by a mobile piston.
Note that for an ideal gas, the definition (3.21) taken at v q constant yields π/θ = p/T .
Assuming the form (3.23) for the pressure tensor, we may write explicitly P when q is directed
along the y axis; it is easy to see that Pxx = Pzz = π < p, and Pyy = 3p − 2π > p. Since
for an ideal gas the components of the pressure tensor are related to the second moments of
the velocity, we have
h 21 mνx νx i = h 21 mνz νz i = 21 kB θ < 21 kB T
h 21 mνy νy i = 21 kB (3T − 2θ ) > 21 kB T
(3.24)
wherein νx , νy , νz denote the velocity components, and brackets stand for mean values. Two
points are worth mentioning: (1) the average molecular kinetic energy h 21 mν 2 i is given by
3
k T , in agreement with the definition of T in kinetic theory; (2) out of equilibrium, the
2 B
equipartition theorem, implying that the average kinetic energy along the three axes is the
same, is no longer valid: the average energy is lower in the directions orthogonal to the heat
flux. This consequence could be checked, for instance, by analysing the Doppler broadening of
emission lines in excited rarefied gases along the direction of the heat flux and perpendicularly
to it (Camacho and Jou 1995).
These considerations remind us that thermometry is not a trivial subject. Since the mean
kinetic molecular energy along the direction x is not the same as along the direction y, a
thermometer will indicate different temperature according to its relative position with respect
to the direction of q . This is not surprising, because in non-equilibrium there is no longer
equipartition and, therefore, thermometers which are sensitive to different degrees of freedom
will indicate different values for the temperature. For instance, if we have a mixture of
matter and radiation at different temperatures, a completely reflecting thermometer will be
sensitive only to matter, and will indicate the temperature of matter, whereas a thermometer
with perfectly black walls will give a value which is the average temperature of the mixture.
Analogous modifications also appear in the equation of state for the chemical potential, and
have important consequences in several phenomena. Let us mention a shift of the critical point
and the coexistence lines in the phase diagram of polymer solutions or liquids under a shear
rate (Eu and Ohr 1993, Jou et al 1995) or the displacement of the chemical equilibrium under
flow or heat flux (Lebon et al 1993, Nettleton and Torrisi 1991, Nettleton 1988b, 1996b, d).
Some of these examples will be explicitly analysed in section 11.3. Let us finally mention that
thermodynamical equations of state in non-equilibrium states have also been discussed from a
statistical mechanical formalism based on the projection operator techniques (Taniguchi 1997).
3.3. Stability conditions
Non-equilibrium equations of state modify the stability conditions, which, for systems not too
far from equilibrium, are related to the sign of the second differential of the entropy. As a
simple illustration, consider heat conduction in a rigid solid for which s = s(u, q ). The second
differential of the generalized entropy is given by (Criado-Sancho and Llebot 1993b)
1
1 2 ∂ 2α
∂α
2
+ q
(3.25)
(δu)2 − αδ q · δ q − 2q0 δu · δ q
δ s=−
cv T 2 2 0 ∂u2
∂u
where α = τ1 v/λT 2 , q0 is the reference value of the heat flux, and δ q the perturbation. If
v, τ1 and λ are constant, one has ∂α/∂u = −2α(cv T )−1 and ∂ 2 α/∂u2 = 6α(cv T )−2 . The
expression for δ 2 s is negative definite only for values of |q | less than a critical value |qcrit |
given by |qcrit | = ρuU , with U = (χ/τ )1/2 the maximum speed of thermal waves. Therefore,
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D Jou et al
thermodynamic stability is only guaranteed when the heat flux is lower than this critical value.
When this condition is satisfied, one avoids an unsatisfactory property of the telegrapher
equation, namely, that an initial absolute temperature profile which is positive everywhere
may take negative values during some time intervals (Criado-Sancho and Llebot 1993b); this
unpleasant result is not found with parabolic diffusion equations, where the positive character
of the temperature field is always respected. We, in fact, consider this limitation on the value
of the heat flux as a consistency criterion for the models of heat transport in the presence of
a finite speed. For instance, in section 14.2 we comment on the role of flux limiters in the
description of nonlinear heat transport.
A more general treatment with both q and P ν as variables may be found in Jou et al (1996a,
chapter 2). Chen (1991) has studied the relation between dynamical and thermodynamical
stability in EIT, and has shown that the thermodynamic equilibrium state is thermodynamically
stable if it is asymptotically dynamically stable.
4. Comparison with other non-equilibrium theories
In section 2 we proposed EIT as a generalization of classical irreversible thermodynamics, and
we have also presented it in the framework of rational thermodynamics. Here, we compare
EIT with two other thermodynamic theories which are also more general than CIT, namely
theories with internal variables and theories including, as additional variables, second moments
of fluctuations of classical variables.
4.1. Theories with internal variables
Internal-variable theories (IVT) (e.g. Ciancio et al 1990, Ciancio and Verhás 1991, Maugin
and Muschik 1994) include additional variables which allow for a more detailed description
of the system than the local-equilibrium approach; they have been successfully applied in
rheology (Verhás 1997) and in electrodynamics of continua (Eringen and Maugin 1990). Some
approaches provide a link between the apparently very dissimilar formulations of rational
thermodynamics and of classical irreversible thermodynamics, by defining an accompanying
equilibrium state that is different from the actual non-equilibrium state. This accompanying
equilibrium state is the adiabatic projection of the actual state on the manifold of equilibrium
states (Kestin 1990, 1993, Muschik 1990a, b, 1993).
These theories have several connections with EIT, since they introduce more variables
and more evolution equations. On the other hand, while EIT uses the macroscopic fluxes as
variables, in these theories the internal variables are either left unidentified, or selected on the
basis of microscopic analyses.
Consider, for example, heat conduction, on which we have already focused our attention.
Instead of directly assuming, as in EIT, that q is an independent variable, one supposes the
existence of a vectorial internal variable g , for the moment not identified, and of the entropy s
depending on the internal energy u and g . The corresponding Gibbs equation is
ds = θ −1 du − vα 0 g · dg
(4.1)
with θ the generalized temperature, v the specific volume and α 0 a scalar field. After
differentiation and using the energy balance equation, (4.1) becomes
ρ ṡ + ∇ · (θ −1 q ) = q · ∇θ −1 − α 0 g · ġ .
(4.2)
In the linear domain, the constitutive equations would take the form
q = L11 ∇θ −1 + L12 (−α 0 g )
ġ = L21 ∇θ −1 + L22 (−α 0 g )
(4.3)
Extended irreversible thermodynamics revisited (1988–98)
1059
with Lij transport coefficients which satisfy the requirements set by the positiveness of the
entropy production and the Onsager–Casimir reciprocity relation (L12 = −L21 , where the
minus sign is due to the fact that g is odd with respect to time-reversal). Combination of the
two equations of (4.3) yields, in the linear regime
q̇ + α 0 L22 q = L11 ∇ θ̇ −1 + α 0 (L22 L11 + L212 )∇θ −1 .
(4.4)
The heat flux now follows an evolution equation more general than the Maxwell–Cattaneo
equation (equation (4.4) is sometimes called a Jeffreys equation, in analogy with the well
known rheological model, or a double-lag equation, see (10.48)). Although it is in many
respects a very useful equation (see section 10, which is devoted to a detailed analysis of nonlocal effects in heat conduction), it yields a parabolic differential equation for the temperature
evolution, thus implying an infinite speed of propagation. It may be observed that the variable
g does not appear explicitly in (4.4), and therefore it is not necessary to identify it, as one is
interested only in the evolution of q rather than the thermodynamic aspects described by the
Gibbs equation (4.1).
Note that (4.4) reduces to the Maxwell–Cattaneo equation if it is assumed that L11 = 0,
α 0 L22 = τ −1 and α 0 L212 = λθ 2 , with λ the thermal conductivity. In this case, the first
equation (4.3) reduces to q = −α 0 L12 g , i.e. the internal variable g is proportional to the heat
flux, and the Gibbs equation (4.1) simply becomes the Gibbs equation (2.38) of EIT.
Thus, we see that an approach based on a vectorial internal variable (not identified a priori)
only leads to the Maxwell–Cattaneo equation if the internal variable is identified a posteriori
as proportional to the heat flux. If this identification is not made, one obtains a more general
evolution equation for q , but one which does not overcome the problem of infinite speed of
propagation.
An analogous situation is found in the analysis of viscous effects. Consider, for example,
the study of polymers; in IVT it is common to introduce by means of supplementary variables
the geometrical configuration of the macromolecular chains (the conformation tensor defined
as C = hRRi, with R the vector going from the first to the last monomer in the chain), in
contrast to EIT, where the viscous pressure tensor is taken as an extra variable. The choice of
variables depends on the experimental measurements: if one controls the shear pressure, its
choice as an independent variable, as in EIT, is more convenient, whereas if a more microscopic
understanding is desired, the conformation tensor is more useful. In some situations (see, for
example, (11.18)), there exists a linear relation between C and P ν so that both theories lead
to similar results.
The main differences between IVT and EIT can be summarized as follows. (1) In IVT,
the evolution equations generally take the form of pure relaxational equations without the
presence of a divergence term. In EIT, the field variables obey balance equations wherein the
fluxes of these variables appear in a rather natural way. (2) Another difference between IVT
and EIT is that the internal variables usually depict some specific microscopic features of the
system, as in the aforementioned example of macromolecules. In EIT, one selects invariably
q , P ν , . . . as variables whatever the nature of the particles composing the system, whether
monatomic, polyatomic with or without internal degrees of freedom. (3) Note, finally, that the
flux variables, such as q , P ν , . . . , are controllable from the external world, which is not true
for internal variables.
4.2. Theories with fluctuations
In some generalized thermodynamic theories (Jaworski 1981, Keizer 1987, Nicolis 1995) one
selects as complementary variables the second moments of the fluctuations instead of the
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D Jou et al
fluxes. The motivation behind such a choice is that, near bifurcations or critical points, the
fluctuations are very much increased and, therefore, not only the mean values of the variables
but also their deviations from their average values become significant. Rather than the classical
thermodynamic potentials, stochastic potentials are introduced. The aim of such theories is to
evaluate the probability density ρ̃(X, t) to be in a state X at time t, by solving the evolution
equation for ρ̃, which takes usually the form of a master or a Fokker–Planck equation. This
can be achieved provided there exist appropriate space and time windows where the dynamics
of fluctuations in the corresponding coarse-grained description is Markovian. The solutions
of such equations are usually written as (Nicolis 1995)
1
X
X
, t + U1
,t + ···
(4.5)
ρ̃(X, t) = C exp[−U (X, t)] = c exp − U0



where  is the size of the system. The dominant term U0 is the so-called stochastic potential,
and it satisfies the Lyapunov property
dU0 (x(r , t))
60
(4.6)
dt
where x(r , t) denotes the solution of the macroscopic deterministic equations. The
macroscopic solution corresponds to the most probable state of the system, rather than to
the mean value. In general, the stochastic potential may have several extrema: for instance,
a bifurcation is reflected by a switch from a single-maximum function to a several-maxima
function or to a fractal function, depending on the attractor. The master equation or the
Fokker–Planck equation has only one solution at bifurcations, whereas the macroscopic
equations (related to the maxima) have several solutions. A thermodynamic analysis relating
the stochastic potential to an information theoretic entropy has been performed (Nicolis 1995).
EIT has no direct link with this point of view, since it deals with macroscopic equations rather
than with a master or a Fokker–Planck equation. Furthermore, up to now bifurcations have
not been discussed in the context of EIT.
A similar point of view was adopted by Keizer (1987) starting from a canonical
representation for the rates of the elementary processes of change of the molecular quantities.
This representation yields explicit expressions for the second moments of the fluctuations
of the corresponding macroscopic quantities, which may have, in general, non-equilibrium
contributions. To define the macroscopic non-equilibrium entropy, Keizer uses Einstein’s
formula relating the probability of fluctuations to the second differential of the entropy, in such a
way that the non-equilibrium contributions to the second moments of the fluctuations determine
the non-equilibrium contributions to the entropy. Keizer’s formulation has, in common with
EIT, its interest in a non-equilibrium entropy. However, it differs from EIT in two basic
ways: it starts from a molecular basis rather than building directly a macroscopic theory, and
it uses the non-equilibrium fluctuations to define the entropy instead of starting from a nonequilibrium entropy to evaluate the fluctuations. However, it is tempting to establish some
indirect connection between both approaches. Indeed, in EIT the fluxes are responsible for
the deviation of the second moments of the fluctuations of the classical variables with respect
to their equilibrium values. Therefore, it is, in principle, possible to write the non-equilibrium
part of the extended entropy in terms of the non-equilibrium fluctuations, in analogy with
Keizer’s theory.
To do that, consider the simple problem of heat conduction. According to EIT, the
probability W of the fluctuations of the energy in a non-equilibrium state for a fixed value
of the heat flux is given by
W (δu) ≈ exp{(M/2kB )[(θ −1 )u (δu)2 ]}
(4.7)
Extended irreversible thermodynamics revisited (1988–98)
1061
where subscript u stands for the partial derivative with respect to the internal energy u and M
the total mass of the system. Here we only consider energy fluctuations, but one could without
difficulty extend the analysis to more general situations (see Jou et al 1996a, chapter 4). In
view of expression (3.14) for θ, and after derivation with respect to u, it is easily checked that
hδuδui = hδuδuieq (1 + Aq 2 )
− 21 cv T 2 ∂ 2 (τ/ρλT 2 )/∂u2 .
(4.8)
Solving (4.8) in terms of q and
where A stands for A =
substituting in the expression (3.17) of entropy, one obtains an expression of s in terms of
the fluctuations of the internal energy, namely
τ
hδuδui
−1 .
(4.9)
s(u, hδuδui) = seq (u) −
2ρλT 2 A hδuδuieq
2
For an ideal monatomic gas, one has
1
hδuδui
s(u, hδuδui) = seq (u) − nkB
−1
8
hδuδuieq
and the corresponding temperature θ takes the form
1
1
1 hδuδui
=
1−
−1 .
θ
T
4 hδuδuieq
(4.10)
(4.11)
This result shows that the non-equilibrium contributions to temperature may equally well be
written in terms of the second moments of the fluctuations of the slow variables or in terms
of the dissipative fluxes as in expression (3.14). In that respect it may be said that there exist
some points of contact with Keizer’s formulation. Nevertheless, it should be kept in mind that
the starting points of both theories are fundamentally different: Keizer (1987) starts essentially
from a master equation for the fluctuations of the variables, and defines the non-equilibrium
entropy by means of the Einstein relation for the probability of fluctuations, while EIT assumes
the existence of a non-equilibrium entropy and takes for granted the Einstein relation (with the
generalized entropy) to obtain the probability of fluctuations.
Again, the choice of the most convenient variables depends on the problem under
consideration. In the analysis of non-equilibrium bifurcations, wherein the fluctuations play
an important role, the second moments approach seems to be well advised, although it is not
incompatible with the use of the fluxes.
5. Hamiltonian and variational formulations
Up to now two kinds of arguments were used to justify the main statements of EIT: first, the
comparison of their consequences with experimental observations; and second, their link with
some microscopic descriptions. Both of these arguments have been widely discussed in the
literature. Our purpose is to offer a new argument supporting these assumptions. The argument
is based on the observation that classical irreversible thermodynamics, hydrodynamics,
rheology, as well as other well established modes of description, like Boltzmann’s kinetic
theory or Maxwell’s electrodynamic equations, possess a so-called generalized Hamiltonian
structure (see Grmela (1986, 1989), Beris and Edwards (1994), Grmela and Öttinger (1997)
and Öttinger and Grmela (1997) for a detailed treatment on this topic). There are other reasons
that militate in favour of a Hamiltonian description. It is very attractive to formulate the
behaviour of a whole class of phenomena in Hamiltonian form because of its conciseness and
its physical content. Indeed the whole set of balance equations is now replaced by one single
relation and, furthermore, the generating functional of each particular problem may generally
be identified with a well-defined physical quantity, such as the energy, the entropy, the Gibbs
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D Jou et al
free energy, etc. Moreover, there exist many elegant results and powerful methods of solutions
typically developed for general Hamiltonian systems which can be of direct use in analysing the
solutions of the basic equations of EIT. Finally, Hamiltonian techniques have a wide range of
validity and provide supplementary restrictions which complement those of the second law as
well as a link between thermodynamics and dynamics at several levels of description, ranging
from the macroscopic to the microscopic (Grmela 1993a–d, Grmela and Jou 1993).
It was shown (Grmela and Lebon 1990, Grmela and Jou 1991, Vázquez and del Rı́o
1996, Grmela et al 1998) that the Hamiltonian structure is preserved when the state of the
basic variables is enlarged in the way advocated in EIT. This connection is important, not only
because it provides further arguments in favour of the internal consistency of EIT but also
because it suggests some restrictions on the nonlinear generalizations of the theory.
5.1. Hamiltonian structure of classical thermo-hydrodynamics
For clarity, we briefly recall the generalized Hamiltonian structure of local-equilibrium
irreversible thermodynamics and also introduce some terminology and notation.
The state variables denoted are the five hydrodynamic fields: mass density ρ(r , t), energy
e(r , t) per unit volume, and momentum u(r , t) related to barycentric velocity by means of
u = ρ v . Note that e(r , t) differs from the internal energy u(r , t) by the kinetic energy
uk uk /2ρ. We first consider the non-dissipative part of the balance laws and demonstrate
their Hamiltonian structure. Here the non-dissipative term is used with the meaning that the
corresponding equation does not contain a production term, and consequently that it is invariant
with respect to time-reversal.
R
It is assumed that there exists an entropy function S(t) = dr s(r , t) such that the entropy
density s(r , t) is a convex function of the variables ρ, e, u and that for an isolated system
dS/dt = 0. It is also supposed that there is a one-to-one transformation between the variables
(ρ, e, u) and (ρ, s, u) for all r and t.
In this section we change from tensorial to component notation, also using Einstein’s
convention for summation over repeated indices because it is customary in the context of
Hamiltonian analyses.
R
Let A(t) = dr a(r , t) denote a sufficiently regular functional of the fields (ρ, s, uk ).
Clearly, one has
Z
δa ∂uk
δa ∂ρ δa ∂s
dA
= dr
+
+
(5.1)
dt
δρ ∂t δs ∂t δuk ∂t
where δ is the Volterra functional differentiation, which reduces to the usual partial derivatives
in the absence of non-locality:
δa
δa
δa
(5.2)
= aρ ,
= as ,
= auk .
δρ(r , t)
δs(r , t)
δuk (r , t)
Here, aρ (= ∂a/∂ρ) denotes the partial derivation of a with respect to ρ, and so on for the
other subscripts. Furthermore, from now on we use a,k for derivation with respect to space
coordinates.
Of course, if we know the time evolution equation for A and require it to hold for all
sufficient regular functionals A, then we can derive the time evolution equations for the fields
ρ, s, uk . If, in addition, the time evolution for A can be cast into the form
dA
= {A, G}
(5.3)
dt
R
where {A, G} is a Poisson bracket and G(t) = dr g(r , t) a generating functional, then the
time evolution equations for ρ, s, uk are said to possess the Hamiltonian structure. We recall
Extended irreversible thermodynamics revisited (1988–98)
1063
that {A, G} is a Poisson bracket if the following three properties are satisfied: (i) {A, G} is a
linear function of the Volterra functional derivatives of A and G with respect to the fields, ρ,
s, uk , respectively; (ii) {A, G} = −{G, A} (antisymmetry); (iii) {A, {B, C}} + {B, {C, A}} +
{C, {A, B}} = 0 (Jacobi identity).
Let us select as
bracket
"our Poisson
# Z
"
#
Z
δa
δg
δg
δg
δa
δg
δa
δa
{A, G} = dr ρ
+
+
+ dr uj
δρ ,k δuk
δρ ,k δuk
δuj ,k δuk
δuj ,k δuk
" #
Z
δa
δg
δg
δa
+
(5.4)
+ dr s
δs ,k δuk
δs ,k δuk
R
and identify the generating functional G with the total energy E(t) = dr e(r , t). We first
show that equation (5.3) is identical to the non-dissipative part of the balance laws of classical
irreversible thermodynamics together with local equilibrium relations among the fields ρ, e, s,
uk and p (p is the local pressure). Performing integration by parts and introducing boundary
conditions that guarantee that all the integrals over the surface bounding the system are zero,
we can rewrite equation (5.3) as
Z
dA
δa
δa
δa
= − dr
(ρeuk ),k + (seuk ),k +
[(uk euj ),j + ρ(eρ ),k + uj (euj ),k + s(es ),k ] .
dt
δρ
δs
δuk
(5.5)
Substituting dA/dt by the right-hand side of (5.1) and requiring that (5.5) holds for any
functional a(r , t), one has
∂ρ
= −(ρeuk ),k
∂t
∂s
(5.6)
= −(seuk ),k
∂t
∂uk
= −(uk euj ),j − ρ(eρ ),k − uj (euj ),k − s(es ),k .
∂t
For expressions (5.6) to represent the non-dissipative parts of the balance laws of classical
irreversible thermodynamics one has to make the following identifications:
1
euk = uk = νk
(5.7)
ρ
where νk is the velocity field, and
p,k = ρ(eρ ),k + uj (euj ),k + s(es ),k .
(5.8)
Equation (5.8) together with
e,k = eρ ρ,k + euj (uj ),k + es s,k
(5.9)
implies p = −e + ses + ρeρ + uk euk which is nothing but the Euler local equilibrium relation.
Now we turn our attention to the dissipative part of the evolution equations. For the sake
of simplicity, from now on we limit the analysis to an incompressible fluid at rest. It is then
an easy task to see that the equations of classical irreversible thermodynamics are recovered
at the condition to replace equation (5.3) by
dA
= {A, E} + [A, S]
(5.10)
dt
where [A, S] stands for the so-called dissipative Ginzburg–Landau bracket (Grmela and
Öttinger 1997)
Z
δ8
δa
[A, S] = − dr
(5.11)
δx δ(∂S/∂x)
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D Jou et al
where x stands for the set of variables ρ, e, s, uk , . . . , while 8 is a so-called dissipative potential
required to obey the following properties: (i) 8(0) = 0; (ii) 8 reaches its minimum at zero;
(iii) −8 is convex in the neighbourhood of zero. These properties are met if dS/dt > 0.
The form of the dissipation bracket [A, S] (which is not a Poisson bracket) that includes the
viscosity may be found in Grmela’s paper (1989) to which the reader is referred for more
details. In particular, the momentum equation (5.6c) will contain in the right-hand side an
additional viscous term so that the Navier–Stokes equation will be recovered.
5.2. Hamiltonian structure of extended irreversible thermodynamics
We proceed now to the identification of the Hamiltonian structure of EIT when we select
as basic variables e(r , t) (energy), qk (r , t) (heat flux) and Pkl (r , t) (the symmetric pressure
tensor). The objective is to search for a generating functional G, a Poisson bracket {, } and a
dissipation bracket [, ], all defined in the state space composed by the fields e(r , t), qk (r , t),
Pkl (r , t), such that: (i) the generating functional has the physical meaning of a free energy;
(ii) equation (5.10), linearized about an equilibrium state, is equivalent to the governing time
evolution equations of EIT. Global motion of the system is neglected (uk = 0) so that e(r , t)
can be identified with the internal energy.
5.2.1. The non-dissipative time evolution equations.
entropy is conserved and one has in local form
In the absence of dissipation, the total
∂s
s
(5.12)
= −Jk,k
∂t
where the entropy flux Jks remains unspecified at this stage of the analysis. Let A denote a
sufficiently regular functional of the state variables; its time derivative can be written as
Z
δa ∂Pkl
δa ∂s δa ∂qk
dA
= dr
+
+
.
(5.13)
dt
δs ∂t δqk ∂t
∂Pkl ∂t
As stated earlier, we say that the equations governing the time evolution of s, qk and Pkl exhibit
a Hamiltonian structure when they may be expressed in the form (5.3) with the total energy E
serving as a generating functional. By defining the Poisson bracket by
" #
Z
δe
δe
δa
δa
−
{A, E} = dr µ1
δs ,k δqk
δs ,k δqk
"
#
Z
δe
δe
δa
δa
−
(5.14)
+ dr µ2
δqk ,l δPlk
δqk ,l δPlk
then the time evolution equations for s, qk and Pkl implied by (5.3) and (5.14) are of the
Maxwell–Cattaneo type. The quantities µ1 and µ2 in (5.14) are constant coefficients and can
easily be related to the coefficients appearing in extended irreversible thermodynamics. It is
checked that {A, E} defined by expression (5.14) is truly a Poisson bracket.
Performing integration by parts, equation (5.14) can be cast into the form
"
"
# Z
#
Z
δe
δe
δe
δa
δa
−µ1
+ dr
−µ1
− µ2
{A, E} = dr
δs
δqk ,k
δqk
δs ,k
δPkl ,l
"
#
Z
δe
δa
−µ2
.
(5.15)
+ dr
δPkl
δqk ,l
Extended irreversible thermodynamics revisited (1988–98)
1065
It follows from expression (5.15) that the evolution equation (5.3) for A can be written as
Z
δa ∂Pkl
δa ∂s δa ∂qk
+
+
dr
δs ∂t δqk ∂t
δPkl ∂t
Z
δa
δa
δa
[−µ1 (eqk ),k ] +
[−µ1 (es ),k − µ2 (ePkl ),l ] +
[−µ2 (eqk ),l ] .
=
dr
δs
δqk
δPkl
(5.16)
Since relation (5.16) must hold for all A, by identification of the coefficients of δa/δs, δa/δqk
and δa/δPkl respectively, it is inferred that
∂s
(5.17)
= −µ1 (eqk ),k
∂t
∂qk
= −µ1 (es ),k − µ2 (ePkl ),l
(5.18)
∂t
∂Plk
= −µ2 (eqk ),l .
(5.19)
∂t
Taking into account the generalized Gibbs equation
τ2
τ1
(5.20)
qk dqk +
Pkl dPkl
de = T ds +
λT
2η
(5.17)–(5.19) can be given the forms
1
∂s
(5.21)
=−
qk
∂t
T
,k
∂qk
τ1
(5.22)
= −λT,k + β 00 T 2 λPkl,l
∂t
∂Plk
(5.23)
= 2β 00 ηT qk,l
τ2
∂t
which coincide with the linearized Maxwell–Cattaneo equations (2.34) and (2.36) for systems
at rest, at the condition to identify µ1 and µ2 as
λ
2β 00 ηλT 2
,
µ2 = −
(5.24)
τ1
τ1 τ2
where β 00 is the coefficient appearing in the expression for the non-classical part of the entropy
flux in EIT (see (2.15)). Note also that (5.21) is identical to the entropy conservation law (5.12)
with the entropy flux given by qk /T .
In order to introduce in the next section the dissipative effects, it is convenient to use e
rather than s as independent variable. This is easily done by using the transformation laws
δ
δ
δ
δ
δ
→ se−1 ,
→
− se−1 sqk
(5.25)
δs
δe
δqk
δqk
δe
δ
δ
δ
→
− se−1 sPkl .
(5.26)
δPkl
δPkl
δe
We are then led to another expression of the Poisson bracket (5.14) (for details, see Grmela
and Lebon 1990) leading to the following evolution equations for e, qk and Pkl :
∂e
(5.27)
= µ1 (se−2 sqk ),k − µ2 (se−2 sPkl sql ),k
∂t
∂qk
= µ1 (se−1 ),k + µ2 (se−1 sPkl ),l
(5.28)
∂t
∂Pkl
= µ2 (se−1 sqk ),l .
(5.29)
∂t
µ1 =
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D Jou et al
By taking for µ1 and µ2 the same values as in (5.24), one finds that (5.28) and (5.29) reduce
to the same evolution equations (5.22) and (5.23) as before. Since the entropy is now the
generating function, it is automatically conserved, as may be directly checked.
5.2.2. Dissipative evolution. When dissipation is present, energy is still conserved but entropy
grows in the course of time with a non-negative rate of production σ s > 0. Dissipation is
incorporated in equations (5.28) and (5.29) by following the same procedure as in section 5.1.
One introduces a dissipative potential 8, which is a real-valued functional of sqk , sPkl ; 8 is
assumed to comply with the same conditions of minimum and convexity as presented at the
end of section 5.1. As a consequence of these properties, it is suggested to relate 8 to the rate
of entropy production by means of
δ8
δ8
+ sPkl
.
(5.30)
σ s = sqk
δsqk
δsPkl
The bracket formulation in the presence of dissipation is now modified and it takes the
form (5.10), wherein the dissipative contribution [A, S] is such that
Z
1
δ8
δ8
dr aqk
[A, S] =
+ aPkl
.
(5.31)
2
δsqk
δsPkl
It is directly checked that the evolution equations implied by (5.10), (5.14) and (5.31) can be
written as
∂e
= µ1 (se−2 sqk ),k − µ2 (se−2 sPkl sql ),k
(5.32)
∂t
δ8
∂qk
= µ1 (se−1 ),k + µ2 (se−1 sPkl ),l +
(5.33)
∂t
δsqk
∂Pkl
δ8
= µ2 (se−1 sqk ),l +
.
(5.34)
∂t
δsPkl
In the case where 8 is a quadratic function of sqk and sPkl , a particular form for 8 is
8=
ηT
1 λT 2
sqk sqk + 2 sPkl sPkl .
2
2 τ1
τ2
(5.35)
Relations (5.33) and (5.34) are exactly the evolution equations derived in the linear version of
EIT by taking for s the definition introduced in EIT as (5.20). It must also be noted that terms
in νk,j are absent here as a consequence of the assumption that the system under consideration
is at rest. This restriction is lifted in a recent work by Lebon and Grmela (1996) and Grmela
et al (1998), where non-local terms and nonlinear effects are considered.
Here, we have shown that, in common with other modes of description, the basic equations
of EIT share the property of possessing a Hamiltonian structure provided that an adequate
expression is given for the generalized entropy, which is seen to have the form advocated by
EIT.
5.3. Variational formulations
Variational formulations may be useful because they provide numerical techniques (Ritz’s,
Galerkin’s, etc) to obtain approximate solutions; in addition, they may also convey a physical
meaning as they generally involve a single scalar quantity to be made extremum. Except
in some particular situations, fluid flows with dissipation cannot be described by means of
variational principles; this is the reason why so-called ‘restricted’ variational principles have
been proposed (e.g. Lebon 1980) wherein only a limited number of quantities are submitted to
Extended irreversible thermodynamics revisited (1988–98)
1067
variation. Examples of ‘restricted’ principles in EIT are those of Vázquez and del Rı́o (1990,
1993), del Rı́o et al (1992b) and Chen (1991). This procedure was widely used in the period
before 1988: by Bhattacharyia within the wave approach of thermodynamics; by Nettleton,
who introduced a Lagrangian formulation of EIT along lines similar to those of Onsager; by
Sieniutycz, which used a relaxation factor in a functional to derive hyperbolic equations of
EIT and by Eu using generalized equations of state (see Jou et al (1988a) for bibliographical
references).
The Onsager–Machlup formalism and the minimum entropy production principle of
Prigogine were used by Ichiyanagi (1995a) to check the consistency and the range of validity of
EIT (see also Eu and Ichiyanagi 1996). Another application of the minimum entropy production
principle in EIT may be found in Lebon and Dauby (1990) for the Guyer–Krumhansl equation
for heat transport (see section 10.5). Ortiz de Zárate and Pérez-Cordón (1997) have explored
the sensitivity of the entropy production with respect to the initial conditions for the hyperbolic
heat conduction.
Nyiri (1991) has discussed the construction of potentials and variational principles in thermodynamics and has illustrated his analysis with an example from the Gyarmati’s wave approach to thermodynamics. Gambar et al (1991) and Gambar and Markus (1994) have derived
a variational principle containing the balance and constitutive equations for the wave approach
in convective systems. A rather different approach was proposed by Anthony (1989) who introduced a complex-valued function for the ‘thermal field’ whose square modulus is equal to the
temperature; the phase is considered as a new degree of freedom which in some circumstances
allows us to recover the hyperbolic heat transport equation (see also Sievers and Anthony 1996).
Sieniutycz (1994) and Sieniutycz and Berry (1989, 1991, 1992, 1993) have provided an
approach based on Noether’s theorem which leads, starting from a suitable Lagrangian, to
the explicit form of the energy–momentum tensor and the balance equations in fluids. These
authors take as independent variables mass density, entropy density, mass flux (momentum
density) and entropy flux (rather than the heat flux, because it may have some advantages in
relativistic formulations, where there is a well-defined four-vector for the entropy, in contrast
with the heat flux). Sieniutycz (1994) defines a kinetic potential as
(5.36)
L(ρ, s, Jk , Jks ) = 21 ρ −1 Jk Jk + 21 ρ −1 gJks Jks − ρueq (ρ, s) − ρ9
2
2
where g takes account of the thermal inertia, and whose value is g = 2m /5kB for ideal gases,
and 9 is the potential energy per unit mass due to external fields. Then, it is postulated that
the action integral of L over the spacetime must be stationary under the constraints resulting
from the entropy and mass balances. These conditions are introduced with auxiliary Lagrange
multipliers in such a way that, after some integrations by parts, the integral to be extremum is
Z Z
∂φ
∂η
s
L− ρ
+ Jk φ,k + ρs
+ (sJk + Jk )η,k
A = 3 dr dt =
∂t
∂t
∂α
(5.37)
+ Jk α,k
dr dt.
+λ ρ
∂t
Here, φ may be identified with the velocity potential when the flow is irrotational and the
convective derivative of η is a temperature; α stands for the Lagrangian coordinate which
ensures the identity of the fluid particle in the Lagrangian description of the fluid.
By postulating the invariance of the action A with respect to translations in time and space
and invoking the well known Noether’s theorem one may obtain the energy–momentum tensor
Gαβ (α, β = 1, 2, 3, 4). The corresponding stress tensor Gij (i, j = 1, 2, 3) is (Sieniutycz
1994)
1
1 ∂g
(5.38)
Jks Jks δij
Gij = ρui uj − ρ −1 gJis Jjs − peq (ρ, s) −
2
2 ∂ρ s
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D Jou et al
which exhibits the dependence with respect to thermal effects as already found in (3.22).
Analogous calculations may be performed to obtain the form of the energy flux Gj 4 . It may
be verified that the Lagrangian and the energy–momentum tensor remain invariant when the
heat flux qk or the thermal momentum defined as Iks = ∂L/∂Jks = ρ −1 gJks , are used instead
of the entropy flux as the independent variable.
B. Microscopic foundations
This section is devoted to the microscopic foundations of extended irreversible
thermodynamics; in particular it is shown that EIT has gained microscopic confirmations
from information theory, the kinetic theory of gases, stochastic phenomena, and computer
simulations. Of course, comparison between microscopic results and thermodynamics is not
only helpful for thermodynamics, as it emerges in explicit expressions for the equations of
state and transport coefficients, but also for microscopic theories, for which thermodynamic
relations offer a check of consistency and a way to obtain maximum information from the
calculation of only a few quantities.
6. Information theory
The bases of EIT are rather well established if one does not go beyond the second-order
approximation in the fluxes. Inclusion of higher-order nonlinear terms makes the results
non-unique and raises several problems. In that respect, information theory is useful as it is
not limited to second-order terms (Vasconcellos et al 1991a, b, 1995a–d, Garcı́a-Colı́n et al
1994, Nettleton 1988c, 1989, 1990a, b, Nettleton and Freidkin 1989, Ichiyanagi 1995b, 1996).
The choice of variables is an important problem in the context of information theory but the
formalism by itself does not allow one to specify the relevant variables, which must be selected
from experimental or other theoretical considerations.
The point of view of kinetic theory, requiring that in a steady state the distribution function
should not depend on time, may be too restrictive, as it implies that all possible macroscopic
variables will be time independent. However, from a more general point of view one could
define a steady state by the condition that only the measured variables do not depend on time
while the other non-observed variables could be allowed to be time dependent. For instance,
in the kinetic theory it would be possible for a given number of moments (say, the first 13
moments) not to depend on time, while some (or all) of the higher-order moments would be a
function of the time, according to the boundary conditions to be satisfied by those moments.
This lack of uniqueness in the distribution function implies also a lack of uniqueness in the
macroscopic entropy. A possible way to define a unique non-equilibrium entropy would be to
assign it the maximum possible value compatible with the constraints imposed by observations;
that is precisely the way suggested by the information-theoretical approach.
Consider a system of N particles characterized by their positions and momenta, µ0 =
{r1 , p1 , . . . , rN , pN } and assume that the local mean values hAi i of a set of extensive
observables Ai (µ0 ) are known. The problem is to obtain the probability density fN (µ0 ) which
maximizes the information about the system compatible with some measured quantities. In
other words, the problem amounts to determining the probability density which maximizes the
global entropy S
Z
3N
−1
S = −kB (h N!)
(6.1)
fN (µ0 ) ln fN (µ0 ) d0N
Extended irreversible thermodynamics revisited (1988–98)
subject to the constraints
(h3N N!)−1
Z
fN (µ0 )Ai (µ0 ) d0N = hAi i
(i = 1, 2, . . .)
1069
(6.2)
for the observables Ai whose value is kept fixed. Here d0N = dr1 dp1 . . . drN dpN is the
volume element in the phase space and h is Planck’s constant. Here, we shall focus our
attention on steady states, for which the mean values of the relevant variables do not depend
on time.
Maximization of S subject to the constraints (6.2) implies maximizing the quantity
Z X
0
−kB
λi (r ) · Ai (µ ) d0N
(6.3)
fN ln fN + fN λ0 + fN
i
where the λi (r ) are the Lagrange multipliers corresponding to the constraints (6.2). The dot
between λi and Ai indicates a scalar product. Finally, λ0 is the Lagrange multiplier accounting
for normalization.
Expression (6.3) is an extremum under the condition that fN satisfies
X
δ
λi (r ) · Ai (µ0 ) = 0
(6.4)
fN ln fN + fN λ0 + fN
δfN
i
from which it follows that
X
λi · Ai (µ0 )
fN = Z −1 exp −
(6.5)
i
where we have written 1 + λ0 = ln Z, with the partition function Z defined as
X
Z
Z = (h3N N!)−1 exp −
λi · Ai (µ0 ) d0N .
(6.6)
i
The Lagrange multipliers are obtained from the constraints (6.2). The latter may be written
in the compact form
−
∂ ln Z
= hAi i
∂λi
(6.7)
as follows from (6.2) and the definition (6.6) of Z. Introduction of the distribution density (6.5)
in the definition (6.1) for the entropy yields
X
λi · h A i i .
(6.8)
S = kB ln Z +
i
In equilibrium, when a system is in contact with a thermal bath in such a way that
its average energy is known, one takes as an observable quantity the Hamiltonian of the
system, A(µ0 ) = H(µ0 ). The corresponding distribution function (6.5) is that of the canonical
ensemble. If, in addition to the energy, the number of particles is chosen as an observable
quantity with a specified mean value, the distribution function takes the form
f = Z −1 exp[−β H(µ0 ) − αN 0 ]
(6.9)
with N 0 the microscopic particle-number operator. To obtain a physical interpretation of the
Lagrange multipliers α(= λ1 ) and β(= λ2 ), one writes the differential expression of (6.8),
namely
X
X
X
dS = kB
λi · dhAi i + kB
hAi i · dλi + kB d ln Z = kB
λi · dhAi i.
(6.10)
i
i
i
1070
D Jou et al
The second equality follows from differentiating theP
expression (6.7) for Z with respect to the
Lagrange multipliers, which results in d ln Z = − i hAi i · dλi . By identification of (6.10)
with the Gibbs equation of classical thermodynamics
dS = T −1 dU − µT −1 dN
(6.11)
one is led to the usual identifications β = (kB T )−1 , and α = −µ(kB T )−1 , with T the absolute
temperature and µ the chemical potential.
In a non-equilibrium state characterized by P ν and q in addition to the classical variables,
one should take into account the constraints on the fluxes. This would lead to a distribution
function of the form
f = Z −1 exp[−β H − αN 0 − λP ν : P ν − λq · q ].
(6.12)
Note that the value of the maximum of the entropy at non-equilibrium steady states (i.e. for
non-vanishing q or P ν ) is lower than its maximum at equilibrium, and therefore, asking for
maximum entropy in the non-equilibrium state is not in contradiction with the fact that the
entropy reaches its absolute maximum at equilibrium states.
It must be stressed that the Lagrange multipliers λP ν and λq have no analogue in classical
thermodynamics and therefore cannot be identified in an equilibrium theory. The Gibbs
equation corresponding to (6.12) is
dS = kB β dU + kB α dN + kB λP ν : dP ν + kB λq · dq .
(6.13)
EIT is the only thermodynamic theory which proposes a Gibbs equation depending on the
viscous pressure and the heat flux. Such an equation has the form (see section 2)
τ2 V
τ1 V
dS = θ −1 dU + πθ −1 dV − µθ −1 dN −
P ν : dP ν −
q · dq .
(6.14)
2ηT
λT 2
We neglect for simplicity the bulk viscous effects (i.e. p ν = 0), so that in what follows P ν
must be understood as the shear pressure tensor.
Comparison of (6.13) with (6.14) allows us to identify the Lagrange multipliers as
1
µ
,
β=
,
(6.15)
α=−
kB θ
kB θ
τ2 V
τ1 V
P ν , λq = −
q.
(6.16)
λP ν = −
2kB ηT
kB λT 2
Note that the exponential form of (6.12) is analogous to the canonical form proposed by Eu
(see (7.48)) as the basis of a non-equilibrium ensemble (Eu 1995c, 1996). The relation between
Eu’s theory and that of maximum entropy has been a topic of specialized debate (Banach 1989,
Eu 1992).
In this section we have limited our considerations to stationary states. Dynamical equations
have been derived by several methods, such as those proposed by Robertson (1967) and Zubarev
(1974) or the non-equilibrium statistical operator method (NESOM) (Luzzi and Vasconcellos
1990). The consistency between the maximum-entropy formalism and the H -theorem has
been analysed by several authors (Freidkin and Nettleton 1989, 1990, Banach 1989).
It must be realized that the distribution function (6.5) is not the exact one describing the
system, but an approximate distribution function which gives exact results for the variables
taken as constraints, but not for the other variables. Furthermore, expression (6.5) lacks
information about the dissipation and the microscopic dynamics of the system, i.e. about the
dynamics of the variables not included as constraints in the maximum-entropy description.
A way out of this is to follow the MaxEnt-NESOM approach (Luzzi and Vasconcellos 1990)
where one introduces a statistical operator (density matrix) of the Zubarev form
ρκ (t) = ρ(t, 0) + ρκ0 (t).
(6.17)
Extended irreversible thermodynamics revisited (1988–98)
1071
Here, ρ(t, 0) is a ‘coarse-grained’ non-dissipative term which has the Gibbsian-like form given
by (6.5), whereas ρκ0 is given by
Z t
0
ρκ0 (t) = exp κ
dt 0 eκ(t−t ) ln ρ(t 0 , t 0 − t)
(6.18)
−∞
where κ(κ −1 can be considered as a lifetime) is a positive infinitesimal that goes to zero
after the calculation of the averages is performed. The distribution function ρ(t, 0) gives
the thermodynamics of the system in the ‘coarse-grained’ space defined by the macroscopic
variables, whereas the function ρκ0 does not contribute either to the partition function or to the
value of the macroscopic variables. The density ρκ0 describes the microscopic dynamics of the
fast variables which has not been included in the thermodynamic description. The statistical
operator ρκ (t) satisfies a modified form of the Liouville equation with infinitesimal sources
related to κ which break its time-reversal symmetry and introduce dissipation.
NESOM has provided a basis for the derivation of the dynamical equations of EIT in a much
more general form than that reviewed above (Vasconcellos et al 1991a, b, 1995a–d). Several
applications have been treated, such as non-equilibrium semiconductors (Vasconcellos et al
1990, Vasconcellos and Luzzi 1992), second sound (Vasconcellos et al 1994, 1995d, 1996) and
ultrafast motion of optical phonons in photoinjected highly excited plasma in semiconductors
(Algarte et al 1996), where the use of ρκ (t) is shown in full detail, in contrast with our restriction
to ρ(0).
6.1. Ideal gas under heat flux and viscous pressure: linear approximation
As a first application of the maximum entropy formalism to EIT, consider an ideal monatomic
gas out of equilibrium subjected to a heat flux and a viscous pressure. At each position, the
mean values of the particle number density n, the energy density per unit volume ρu, the
momentum density ρ ν , the heat flux q , and the components of the pressure tensor P are
supposed to be known. Since the equilibrium pressure p is fixed by u and n, the independent
knowledge of P is a constraint on the viscous part P ν of the pressure tensor.
The distribution function (6.12) maximizing the entropy for a constant number of particles
(α = 0) has the form
f = Z −1 exp[−β 21 mC 2 − λν · C − λP ν : (mCC − p U ) − λq · 21 mC 2 C ]
(6.19)
where C = c − ν stands for the relative velocity of the particles with respect to the mean
velocity ν . Here we have introduced a new Lagrange multiplier λν in order to ensure the
constraint on the value of the velocity. Expanding the exponential in (6.19) up to first order in
the non-equilibrium terms results in
f = Z −1 exp[−β 21 mC 2 ][1 − λν · C − λP ν : (mCC − p U ) − λq · 21 mC 2 C ].
(6.20)
By requiring that (6.20) leads to the mean values of density n, velocity ν , internal
ρu = 23 nkB T , heat flux q and viscous pressure P ν , it is found that
3/2
m
−1
Z =n
2πkB T
1
β=
kB T
m
1
2m
λν =
q,
λP ν = −
P ν,
λq = −
q.
pkB T
2pkB T
5pkB2 T 2
energy
(6.21)
(6.22)
(6.23)
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D Jou et al
6.2. Ideal gas under shear flow: nonlinear analysis
The simplest example allowing for a nonlinear analysis is the classical ideal gas under shear
(thermal effects are not present) (Banach and Pierarski 1993, Bidar et al 1996) with the
constraints on the internal energy and the viscous pressure tensor. The technique of the
Lagrange multipliers yields the following result for the distribution function which maximizes
the entropy:
X
XX
(β + 2λii )mCi2 +
2λij mCi Cj
(6.24)
f = z−1 exp − 21
i
i
j (>i)
where β and λij (≡ λ ) are the Lagrange multipliers corresponding to the constraints on the
energy and the viscous pressure tensor, respectively, and z is the one-particle partition function.
Explicit integration of the partition function z gives
Pijν
(2π)3/2 V
(6.25)
m3/2 |M |1/2
with |M | the determinant of the matrix
!
β + 2λ11
λ12
λ13
M =
.
(6.26)
λ12
β + 2λ22
λ23
λ13
λ23
β + 2λ33
The condition Tr P ν = 0 implies λ11 + λ22 + λ33 = 0. In the particular case of a system
ν
, corresponding to a plane Couette flow, the
submitted to a fixed shear viscous pressure P12
only non-vanishing Lagrange multipliers are β and λ12 , and one has
z=
(2π)3/2 V 3
(β − βλ212 )−1/2 .
(6.27)
m3/2
The Lagrange multipliers may be obtained in terms of u (as the internal energy is taken per
ν
as
unit volume and particle) and P12
z=
β=
1−y
;
2V u[R 2 + (1 − y)]
λ12 =
3R 2 + 2(1 − y)
2V uR[R 2 + (1 − y)]
(6.28)
ν
with R = P12
/Nu and y = (1 + 3R 2 )1/2 . Near equilibrium, i.e. when λ12 → 0,
ν
expressions (6.28) tend to β = 3/(2V u) and λ12 = −β 2 (V /N )P12
, respectively. Thus, when
ν
P12 = 0 one recovers from (6.24) the standard Maxwell–Boltzmann distribution function and
β is simply β = (kB T )−1 , with T the local-equilibrium temperature.
The entropy takes the form
NkB 27R 2 [R 2 − (y − 1)]2
ln
(6.29)
2
2(y − 1)3
with Seq the equilibrium entropy. The Lagrange multipliers may be identified according
to (6.15) and (6.16) as
τPν
1
τ γ̇
β=
;
λ12 = − 12 =
(6.30)
kB θ
ηkB T
kB T
where η(γ̇ ) is the shear viscosity with γ̇ = ∂ν1 /∂x2 . Notice that we do not have information
either on τ or on η, but only on their ratio. Nevertheless, the above identification of the
Lagrange multiplier allows us to introduce the shear rate γ̇ in the description of the system.
From (6.28) and (6.30), we are now able to identify the viscosity as
S = Seq +
η=−
ν
P12
3 R 2 [R 2 + (1 − y)]
= − η0 2
γ̇
2 R + (2/3)(1 − y)
(6.31)
Extended irreversible thermodynamics revisited (1988–98)
1073
where η0 is the shear viscosity for γ̇ tending to zero. This expression exhibits the phenomenon
of shear-thinning, i.e. the reduction of the viscosity with increasing shear rate. For low values
of R, (6.31) tends to η0 , whereas it tends to η(γ̇ ) = 0 when R → 1. The physical meaning
of the Lagrange multipliers beyond the linear approximation in the fluxes is a topic of active
current research (Zakari and Jou 1995).
6.3. Ideal gas with non-vanishing heat flux: nonlinear analysis
Consider an ideal non-relativistic gas under fixed values of the energy and the heat flux, at zero
speed (a similar analysis for a dense fluid may be found in Nettleton (1988c)). The distribution
function, according to the maximum-entropy formalism, is
f = z−1 exp[−β 21 mC 2 − λq .( 21 mC 2 − 25 β −1 )C ]
(6.32)
wherein the factor 5/(2β) guarantees that the mean speed is zero. This distribution diverges
for high values of the molecular speed, because the operator for the heat flux is odd in the
velocity; to avoid this divergence it is usual to expand (6.32) in powers of λq , as has been said
above. By keeping second-order terms in λq , one obtains
mC 2
5
1
f = z−1 exp −β
C · λq
1−
mC 2 −
2
2
2β
1 1
5 2
mC 2 −
+
(C · λq )(C · λq ) .
(6.33)
2 2
2β
By imposing the conditions on the density, internal energy and heat flux, one obtains for β the
following expression (Domı́nguez and Jou 1995):
β = (kB T )−1 [1 + ( 25 mp2 kB T )q · q ].
(6.34)
The expression for λq coincides with (6.23c): this is not surprising, because both formulations
must agree up to the first order in the fluxes. It is important to stress that the Lagrange multiplier
β is now different from (kB T )−1 and that it depends on the heat flux. Note, furthermore, that
if the heat flux has the direction of the y axis, λq has also the same direction, according to the
second equation of (6.34). Then (6.33) leads to the result that the average value of 21 mCx2 is
the same than that of 21 mCz2 but different from that of 21 mCy2 , in such a way that equipartition
of energy along the three degrees of translational freedom no longer holds, as commented in
section 3.2.
6.4. Ideal relativistic gas with non-vanishing heat flux
As a further illustration, consider a relativistic gas in a non-equilibrium steady state, with
prescribed internal energy U and integrated energy flux J = V q , where V is the volume and
q the energy flux, i.e. the energy transported per unit area and time. The distribution function
maximizing the entropy reads (Ferrer and Jou 1995)
X
X
pi c − λq ·
pi cc
(6.35)
f = Z −1 exp − β
i
i
where pi c is the microscopic expression for the energy of the ith particle and pi cc is the particle
contribution to the energy flow (all particles are supposed to move at the same speed c). In
contrast with the classical gas studied in the previous section, the distribution function (6.35)
does not diverge because the dependence of the energy flux in terms of the momentum is of the
first order, instead of the third order as in the classical gas. This allows one to derive an explicit
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D Jou et al
expression for the partition function without introducing any truncation of the exponential. The
partition function is given by (Ferrer and Jou 1995)
#−2N
"
c2 λ2q
1
8πV N
Z=
.
(6.36)
1− 2
N! β 3 c3 h3
β
The Lagrange multipliers β and λq are determined from the conditions on the mean energy and
the mean energy flux, which can be obtained, according to (6.7), in terms of the derivatives of
ln Z with respect to β and λq . Solving for β and λq one is led respectively to
β=
3N 1
U y0 − 1
(6.37)
and
λq =
where y 0 stands for
3N y 0 − 2
J
J 2 y0 − 1
"
J
y = 4−3
cU
0
(6.38)
2 #1/2
.
(6.39)
It follows from (6.37) and (6.38) that for J = 0 one has λq = 0 and β = (kB T )−1 ; in addition,
one recovers from (6.36) the usual equilibrium expression for the partition function. In general,
for J 6= 0, β 6= (kB T )−1 , and the non-equilibrium temperature θ is given by θ = T (y 0 − 1).
Furthermore, it is seen that β and λq diverge for y 0 = 1, i.e. when J tends to U c. This domain
of validity of J is easy to interpret, because cU is the maximum energy flow which may be
transported: it corresponds to the energy U carried at the maximum possible speed, which is
precisely c.
The entropy is given by
S = kB [ln Z + βU + λq · J ]
(6.40)
and it may be cast in terms of y 0 as
1
S = Seq + NkB ln[ 16
(y 0 − 1)(y 0 + 2)2 ]
(6.41)
which shows that the entropy depends on the energy flux J in a non-quadratic way. Domı́nguezCascante and Faraudo (1996) have shown that (6.35)–(6.38) and (6.41) may be obtained by
performing a Lorentz transformation of the distribution function and the entropy from a system
in equilibrium at rest to a system moving with momentum J c−2 .
It is also easy to obtain the expression of the pressure tensor for the relativistic gas under
a heat flux q ; it is found that
U 1 0
qq
P =
(y − 1)U − (y 0 − 2) 2 .
(6.42)
V 3
q
Notice that, for y 0 = 2, the pressure tensor reduces to P = 13 (U/V )U = p U , namely to
the equilibrium pressure tensor. The term in qq becomes significant when the magnitude
of the energy flux approaches cU (i.e. y 0 → 1) and it plays an important role in radiation
hydrodynamics, where one writes the pressure tensor as (Anile et al 1991, Domı́nguez and
Jou 1995, Domı́nguez-Cascante and Faraudo 1996)
U 1 − χE
3χE − 1 qq
P =
U+
(6.43)
V
2
2
q2
Extended irreversible thermodynamics revisited (1988–98)
1075
χE being the so-called Eddington factor, which is now obtained as a function of y 0 by
comparing (6.42) with (6.43).
If one wants to study electromagnetic radiation, the relevant statistics is that of Bose–
Einstein and one should maximize the following expression for the entropy:
Z
(6.44)
S = −kB (h3N N!)−1 [f ln f − (1 + f ) ln(1 + f )] d0.
This yields for the distribution function
f = [exp(βpc + λq · pcc) − 1]−1 .
(6.45)
Here, we do not consider a fixed number of particles because we are dealing with photons
whose particle number is not fixed. The calculations of β and λq are rather cumbersome
and the final results are (Larecki 1993, Müller and Ruggeri 1993, Domı́nguez-Cascante and
Faraudo 1996, Fort 1997)
1
aV 1/4 (y 0 + 2)1/2
β=
(6.46)
2kB U
(y 0 − 1)3/4
3
1
J
(aV U 3 )1/4 0
(6.47)
λq = −
2kB
(y + 2)1/2 (y 0 − 1)3/4 c2 U 2
where a is the radiation constant coming from the well known expression U = aT 4 V for the
internal energy of radiation at equilibrium. In terms of y 0 , the entropy can be expressed as
3/4
S
2
U
(y 0 − 1)1/4 (y 0 + 2)1/2
(6.48)
= a 1/4
V
3
V
which tends to the expected value S/V = 43 aT 3 at equilibrium (J = 0, i.e. y 0 = 2) and
which vanishes for J → cU . An expression of the non-equilibrium entropy for radiation
hydrodynamics has been derived by Domı́nguez-Cascante (1997) when, instead of the energy,
one imposes the number of photons. An application of the maximum-entropy technique to nonsteady situations in radiation hydrodynamics has been developed by Mao and Eu (1993), Eu and
Mao (1992, 1994), Fort and Llebot (1998), Fort et al (1998), and Mascali and Romano (1997).
6.5. Harmonic chain with non-vanishing heat flux
In a harmonic chain, the phonon mean free path is infinite, so that the energy flux along it
is not proportional to the temperature gradient but to the temperature difference between the
reservoirs located at its ends. To avoid complications associated with the boundary conditions,
Miller and Larson (1979) considered a closed chain forming a ring characterized by the
constraints hHi = U , hJ i = Q, with H the Hamiltonian of the system, J the heat flux
operator, U the mean (internal) energy of the chain, and Q the mean heat flux along the ring.
The system consists of a chain of N particles of mass m connected to their nearest neighbours
by Hookean springs with stiffness κ. It is convenient to introduce dimensionless quantities
with the mass expressed in terms of m, time in units of (m/κ)1/2 , and energy in units of
(h/2π )(κ/m)1/2 . Let qα be the displacement from equilibrium of a particle α(α = 1, . . . , N)
and pα its conjugate momentum.
The Hamiltonian of the system H(q1 , p1 , . . . , qN , pN ) is given by
X
X
H = 21
pα2 + 21
(qα+1 − qα )2 .
(6.49)
α
α
The microscopic operator J (q1 , p1 , . . . , qN , pN ) for the heat flux is
X
J = − 21 N −1
(qα+1 pα − qα pα+1 ).
α
(6.50)
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D Jou et al
From Hamilton’s equations it can be verified that J is a constant of the motion.
The distribution function is given by
f = Z −1 exp[−β H − λq J ]
with the partition function
Z = (hN −1 )−1
(6.51a)
Z
exp[−β H − λq J ] d0N−1 .
(6.51b)
The Lagrange multipliers β and λq may be expressed in terms of U and Q through the
constraints
β=
1 + x2
,
ε(1 − x 2 )
λq = −βN
2x
(1 − x 2 )
(6.52)
with ε = U/N ≡ kB T , being T the local-equilibrium temperature, and x = Q/ε.
The Lagrange multiplier β can be interpreted in terms of a generalized absolute temperature
θ defined as θ = (kB β)−1 ; from (6.52) one has
1 + x2
2x 2
−1
θ −1 = T −1
=
T
1
+
(6.53)
1 − x2
1 − x2
indicating that the generalized temperature θ differs from the usual local-equilibrium
temperature.
In terms of quantities ε and x, the partition function is
Z = [ε(1 − x 2 )]N .
(6.54)
For x = 0 one recovers the usual equilibrium results, whereas for x → 1 both β and λq
diverge. In the thermodynamic limit when N tends to infinity, the entropy per particle turns
out to be
S
= kB [1 + ln ε + ln(1 − x 2 )] = seq + kB ln(1 − x 2 ).
(6.55)
s = lim
N →∞ N
The dependence of this expression with respect to the heat flux Q provides further evidence
of the basic assertion of extended irreversible thermodynamics: that out of equilibrium, the
entropy is a function of the heat flux.
The entropy (6.55) diverges when x → 1, i.e. when the absolute temperature θ tends to zero
(in this limit the heat flux tends to the maximum value). This result is a consequence of classical
statistics rather than quantum statistics. Indeed, when the non-equilibrium temperature θ
becomes lower than the Einstein temperature of the lattice, it is necessary to resort to Bose–
Einstein’s rather than classical statistics (Camacho 1995b); therefore, the results obtained by
Miller and Larson are not valid for a heat flux larger than a given value.
We briefly turn to the quantum description. Consider a harmonic chain with a linearized
dispersion relation ω = c|k|, c being the phonon speed and |k| the magnitude of the wavevector.
A quantum analysis of the system under the constraints of a fixed energy density ε and a fixed
energy flux q yields for the distribution function maximizing the entropy
2
f (k; β, γ ) = [exp(βh̄c|k| + λq h̄c2 |k|) − 1]−1
(6.56)
where β and λq are the Lagrange multipliers and h̄ = h/(2π ).
The entropy behaviour in the quantum limit εD 1 (with εD the Debye energy
εD = h̄cπ/ l) is
kB π 1/2
[(εc + q)1/2 + (εc − q)1/2 ].
(6.57)
s=
2 6h̄
Extended irreversible thermodynamics revisited (1988–98)
The Lagrange multipliers β and λq are then given by
1
1
1 π 1/2
+
β =
2 6h̄
(εc + q)1/2 (εc − q)1/2
1
1
1
+
=
2kB T (1 + x)1/2 (1 − x)1/2
1 π 1/2
1
1
−
λq =
2c 6h̄
(εc + q)1/2
(εc − q)1/2
1
1
1
−
=
2ckB T (1 + x)1/2
(1 − x)1/2
1077
(6.58)
(6.59)
where T (ε) ≡ (6h̄cε/kB2 π)1/2 is the local-equilibrium temperature and x = q/εc. At
equilibrium, q = 0 and consequently λq = 0, β = (kB T )−1 and (6.56) is identical to the
equilibrium Bose–Einstein distribution function. Note that the results (6.52) are not recovered
in the classical limit because, in the present problem, the linearized dispersion ω = c|k| instead
of the exact dispersion relation has been used.
It is interesting to note that the expression for the specific heat at constant heat flux defined
by cq = (∂ε/∂θ)q vanishes in the limit x → 1, i.e. when the heat flux tends to its maximum
value, which corresponds also to the situation when the non-equilibrium absolute temperature
θ tends to zero. This provides an interesting generalization of the third law to non-equilibrium
steady states (Camacho 1995a, b): indeed, in equilibrium, θ coincides with the equilibrium
temperature so that the vanishing of θ means the vanishing of T ; but out of equilibrium, even
at a nonzero value of T , θ and henceforth cq may become zero at sufficiently high values of
the heat flux.
7. Kinetic theory
Historically, the microscopic justifications of EIT found their roots in the kinetic theory of dilute
gases, mainly Grad’s 13-moment model (Grad 1958). However, it turns out that this approach
has no smallness parameter so that it requires one to keep, in principle, an infinite number of
moments in the description. Indeed, for ideal monatomic gases and for radiation, the relaxation
times of the higher-order fluxes are of the same order of magnitude as those of the lower-order
fluxes, such as the heat flux and the viscous pressure tensor; therefore, when the frequency is
sufficiently high, one should include the infinite set of moments. The question is thus under
which conditions we are allowed to use a mesoscopic theory with a reduced number of fluxes
as variables. A way out is to start from an infinite number of moments, and to expand the
corresponding transport coefficients in the form of continued fractions, whose asymptotic form
leads to the definition of effective relaxation times for the classical fluxes (Dedeurwaerdere
et al 1996). This procedure yields a better agreement with experimental observation than
the classical approach utilizing a reduced number of fluxes; moreover, convergence towards
experimental values is faster than when using a perturbative approach with more and more
higher-order fluxes. A description based on a limited number of fluxes can be considered as a
mean-field description where the fluctuations due to all the neglected higher-order fluxes are
ignored.
7.1. Thirteen-moment approach
The non-equilibrium distribution function may be expanded according to
f = feq [1 + φ (1) + φ (2) + · · ·]
(7.1)
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D Jou et al
where φ (1) , φ (2) , . . . are successive approximations, which may be expressed in terms of the
higher-order moments of the velocity distribution function. The function feq is the equilibrium
distribution function either in global or local equilibrium. The quantities ρ = nm, ν , and T
are determined from the first five moments of the distribution function. This imposes on φ (i)
the closure conditions
Z
Z
Z
(i)
(i)
feq φ dc = 0,
feq φ C dc = 0
feq φ (i) C 2 dc = 0
(i = 1, 2, . . .).
(7.2)
When (7.1) is introduced into the Boltzmann expressions of the entropy and the entropy flux
Z
Z
ρs = −kB f (ln f − 1) dc
J s = −kB f (ln f − 1)C dc
(7.3)
one obtains up to second order
Z
ρs = ρseq − 21 kB
and
1
J = q − 21 kB
T
s
feq φ (1)2 dc
(7.4a)
feq φ (1)2 C dc.
(7.4b)
Z
Owing to the restrictions (7.2), φ (2) does not contribute to the entropy up to the second order of
approximation. Furthermore, it follows from the third condition of (7.2) that the bulk viscous
pressure for an ideal monatomic gas vanishes identically.
To evaluate φ (i) two important models, Chapman–Enskog’s and Grad’s, have been
proposed. In the first one (Chapman and Cowling 1970), f is expressed in terms of the
first five moments n, ν , and T and their gradients. Then, φ (1) is proportional to ∇ ν and ∇T ,
while φ (2) contains terms in ∇∇ ν , ∇∇T and so on.
In Grad’s method (Grad 1958), the non-equilibrium distribution function f (r , c, t) is
replaced by the infinite set of variables ρ(r , t), ν (r , t), T (r , t), an (r , t), where an stand for
the successive higher-order moments of the distribution function, chosen in such a way that
they are mutually orthogonal, and they are given by Hermite polynomials. In the 13-moment
approximation, the expansion is limited to all the second-order moments and to some of the
0
third-order moments, those related to the heat flux. Since P ν and q are directly related to the
second- and third-order moments of the velocity distribution function, Grad considers them as
independent variables and therefore he is closer to EIT than Chapman–Enskog’s approach.
In the 13-moment approximation, the distribution function is written as
0
0
2m
5
m
1
2
mC
k
CC : P ν +
−
T
C
·
q
(7.5)
f = feq 1 +
B
2pkB T
2
5pkB2 T 2 2
0
where CC stands for the traceless part of the symmetric tensor CC . This approximation has
received special attention because all the moments used in it have a well-defined and well
known physical meaning. Such an expression has also received a sound justification on the
bases of the information theory (see equation (6.20)). After substitution of (7.5) into (7.4a)
and (7.4b), the expressions for the entropy and entropy flux turn out to be
1 0ν 0ν
m
P :P −
q·q
4pT
5pkB T 2
1
2 0ν
Js = q −
P · q.
T
5pT
ρs = ρseq −
(7.6)
(7.7)
Extended irreversible thermodynamics revisited (1988–98)
1079
These results confirm the hypotheses underlying EIT stating that the entropy may depend on the
dissipative fluxes and that the entropy flux contains extra contributions besides T −1 q . Note,
however, that these expressions are restricted to second-order developments. This explains
that T −1 instead of θ −1 appears in (7.6) and (7.7). Indeed, the difference between T −1 and
θ −1 is of the second order in the fluxes and then this problem is not raised in Grad’s theory.
Therefore, an analysis of the role of the temperature in the entropy flux would require a more
detailed approach than the one presented here (Domı́nguez-Cascante and Jou 1995).
The evolution equations for the fluxes can be obtained by inserting (7.5) into Boltzmann’s
equation. In the 13-moment approximation one is led to (Grad 1958)
0
0
0
0
0
0
(P ν ). = − 45 (∇ q )s − 2p V − ργ P ν − P ν · (∇ ν ) − (∇ ν ) · (P ν )T
0
0
−P ν (∇ · ν ) + 23 [P ν : (∇ ν )]U
(7.8)
and
0
q̇ = −(kB T /m)∇ · P ν − 25 (pkB /m)∇T − 23 ργ q − 75 q · (∇ ν ) − 25 q · (∇ ν )T
0
0
0
− 75 q (∇ · ν ) − 27 (kB /m)P ν · ∇T + ρ −1 P ν · (∇ · P ν )
(7.9)
where the coefficient γ is given in terms of the collision integrals. The above analysis is not
restricted to one-component gases, but may be extended to diffusion flux in a gas mixture
(Khonkin and Orlov 1994).
Although Grad’s 13-moment approach was one of the initial stimuli for extending the
classical theory of irreversible processes, it presents some deficiencies as mentioned earlier,
which is why some formalisms including higher-order moments (even an infinite number of
them) have been developed.
7.2. Higher-order moments
In the first versions of EIT, the extra variables were the fluxes of heat, momentum, and matter.
However, there is no restriction to enlarge the space of variables by including either a finite
number of fluxes or a whole hierarchy of fluxes, namely the flux of the flux, etc. Now, the
question arises whether it is sufficient to base a mesoscopic description exclusively on the
conserved variables plus the usual dissipative fluxes (or a small number of additional fluxes)
or, on the contrary, whether one should include an infinite number of flux variables. Going
back to Grad’s expansion, it should be recalled that it does not appeal to a smallness parameter;
therefore, it is difficult to find any rigorous justification to truncate the expansion at the 13moment level or at any further stage. Although higher-order fluxes may not directly influence
the values of the lower-order fluxes, their presence makes the solution of Boltzmann equation
under a given value of q and P ν non-unique. An infinite number of solutions, depending on
the boundary conditions imposed on the higher-order fluxes, may be found. Usually, these
conditions are selected in such a way that the higher-order Hermite polynomials expressing
higher-order moments vanish. Whether this is fully realistic or only a convenience is not
completely clear. Therefore, the choice of boundary conditions remains an open problem.
Several authors have considered a finite number of moments (Lukacs and Martinás 1988,
Lebon and Cloot 1988, 1989, Velasco and Garcı́a-Colı́n 1992, 1995, Roldughin 1995). Here
we follow the approach by Velasco and Garcı́a-Colı́n (1992, 1995), who have introduced 20and 26-moment expansions. The physical motivation for these generalizations is that they
are the lowest approximations that yield generalized transport coefficients which depend not
only on the frequency (as in the 13-moment approximation) but also on the wavevector. In the
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D Jou et al
20-moment expansion the variables are, besides the usual 13 moments, the reduced third-order
moments defined as
sij k = Sij k − 25 (qi δj k + qj δik + qk δij )
(7.10)
which is a symmetric traceless third-order tensor, with Sij k given by
Z
Sij k = mCi Cj Ck f dc.
(7.11)
Cartesian coordinates will be used in this section. The 26-moment expansion includes
furthermore the symmetric second-order tensor resulting from the internal contraction of the
fourth-order moment, namely
Z
1
(mC 2 − 5kB T )Ci Cj f dc.
(7.12)
Jij =
2
The distribution function in the 26-moment approximation is given by (Velasco and Garcı́aColı́n 1992)
0
0
2m
5
m
1
ν
2
CC : P +
mC − kB T C · q
f = feq 1 +
2pkB T
2
5pkB2 T 2 2
2
2 m
m
7
1
1
kB T
mC 2
−
+
CCC : s +
CC : J −
P
6p kB T
7p kB T
2kB T
2
m
2 2
2
mC
mC
7
mC
5
m
m
−
−3
−
qij kl Hij(4)kl
Jrr +
+
30pkB T kB T 2kB T
2
2kB T
24pkT
2
(7.13)
where H (4) stands for the fourth-order Hermite polynomial.
The entropy in the 26-moment approximation has the form (Velasco and Garcı́a-Colı́n
1992)
1
m 2 2
m
m
1
Pijν Pijν −
q
q
−
s
s
−
Jrr
ρs = ρseq −
i
i
ij
k
ij
k
4pT
5pkB T 2
12pkB T 2
60pT kB T
1
kB T
kB T
m 2
−
Pij
Pij
Jij −
Jij −
(7.14)
14pT kB T
m
m
and the entropy flux is given by
1
2
5
m
m
1
qj Jij −
sij k Pj k −
sij k Jj k
Jis = qi −
T
5pT kB T
14pT
7pT kB T
m
2
Jrr qi .
−
15pT kB T
(7.15)
The first terms on the right-hand side of (7.14) and (7.15) are the same as in the 13-moment
expansion, which is recovered when the higher-order fluxes are assumed to vanish. In the
linear approximation, the evolution equations for the variables are now given by
0
∂Pij
1
8nm0
kB T
4 0
= − [Pij + 2η(∇ ν )ij ] −
Jij −
Pij − (∇ q )ij − ∇k skij
(7.16)
∂t
τp
105kB T
m
5
0
∂qi
1
1
= − (qi + λ∇T ) − ∇i Jrr − ∇j J ij
(7.17)
∂t
τq
3
∂sij k
5kB T 0
2 0
1
(7.18)
= − sij k −
(∇ P )ij k − (∇ J )ij k
∂t
τs
7m
7
Extended irreversible thermodynamics revisited (1988–98)
1081
Table 1. Maximum propagation speed as a function of the number of moments (Weiss and Müller
1995).
Number of moments
νmax /c0
Number of moments
νmax /c0
10
20
56
120
220
680
1.3416
1.8082
2.5750
3.2103
3.7641
4.9295
969
1 330
4 060
7 770
10 660
15 180
5.3363
5.7185
7.2207
8.2202
8.7450
9.3631
4kB T
1
∂Jrr
=−
∇r qr − Jrr
(7.19)
∂t
m
τ0
∂Jij
p kB T 40
2pkB T 0
1
kB T
=−
+
Pij −
(∇ ν )ij −
Pij
Jij −
∂t
m
η
15
m
τj
m
18kB T 0
2kB T
−
(7.20)
(∇ q )ij −
∇k skij .
5m
m
The relaxation times τ0 , τq , τp , τj , τs and the coupling coefficient 0 are related to the collision
integrals which depend themselves on the choice of the molecular model being considered.
Velasco and Garcı́a-Colı́n (1992) have also derived from the above equations the dependence
of the transport coefficients with respect to the frequency and the wavevector. The frequencydependent viscosity and conductivity at k = 0 coincide for the 13-, 20- and 26-moment
expansions, whereas the wavevector-dependent expressions at zero frequency differ in the
three approximations. As a consequence, the values of the speed of propagations of signals
will depend on the degree of approximation. Some criteria to derive approximations of different
orders in EIT have been discussed by Rodrı́guez and López de Haro (1989) and del Rı́o and
López de Haro (1990).
τ , τ = 0.819τp , τ0 = 0.774τp ,
Furthermore, it turns out that τq = 23 τp , τs = 27 τq = 21
4 p j
which exhibits the property that the relaxation times of the higher-order fluxes are of the same
order of magnitude as those of q and P ν , instead of following a hierarchical scheme in which the
higher-order fluxes would have lower and lower relaxation times. This motivates the selection
of an infinite number of higher-order fluxes in the description, at least in principle. Weiss and
Müller (1995) have considered higher-order approaches, and have computed how the speed of
thermal and viscous signals depends on the order of approximation. Table 1 shows their results
of the maximum propagation speed (referred to the Laplace sound speed c0 ) in a Maxwell gas
as a function of the number of moments. These results exhibit the slow convergence of the
speed values, the increase of the speed with the increasing number of moments, and show
clearly that the Grad 13-moment theory does not provide a satisfactory microscopic basis for
the development of EIT, as there is no way out to determine a priori the number of variables to
be taken. On the other hand, a formalism based on a large number of independent variables is
neither convenient from a practical point of view, nor very interesting from a conceptual view.
We therefore prefer to rely on another approach, which includes the whole set (i.e. an infinite
number) of fluxes, but which uses an asymptotic expression to reduce the number of variables
to be kept on the thermodynamic description.
7.3. Infinite number of moments
Two different approaches involving an infinite number of moments will be analysed (Velasco
and Garcı́a-Colı́n 1993, Dedeurwaerdere et al 1996). Both methods are complementary and
1082
D Jou et al
therefore worth analysing. Velasco and Garcı́a-Colı́n (1993) take for the distribution function
f (r , c, t) = feq (r , c, t)
∞
X
1 (r)
a (r , t) ⊗ H (r) (c)
r!
r=0
(7.21)
where a(r) are the reduced orthonormal moments and H (r) the Hermite polynomials of order
r and the symbol ⊗ denotes the contraction of the corresponding tensors. The linearized set
of evolution equations for the variables a(r) are (for r > 3) of the form
∞
X
1
kB T 1/2
da(r)
[∇ · a(r+1) + ∇ a(r−1) ] = −n
(7.22)
+
{H (s) , H (r) } ⊗ a(s)
dt
m
s!
s=1
where the collision brackets {F, G} stand for
Z
1
{F, G} = 2 F (c)[G(c0 ) + G(c01 ) − G(c1 ) − G(c)]feq (c)feq (c1 )gσ (g, ) d dc dc1
n
(7.23)
with g the modulus of the relative velocity c − c1 , σ (g, ) the collision cross section and 
the solid angle. It may be shown that for F = G these collision brackets are always positive.
The entropy, the entropy flux and the entropy production are respectively given by
∞
nkB X
1 (r)
a (r , t) ⊗ a(r) (r , t)
2 r=1 r!
∞
1
nkB kB T 1/2 X
1 (r)
= q−
a ⊗ (a(r+1) + a(r−1) )
T
2
m
r!
r=1
ρs(r , t) = ρseq (r , t) −
(7.24)
J (s)
(7.25)
σ s = n2 kB
∞
X
1 (r) (p)
a a ⊗ {H (r) , H (p) }.
r!p!
r,p=1
(7.26)
It is seen that: (i) the correction to the equilibrium entropy is quadratic in terms of the extra
variables; (ii) the entropy flux is not simply q /T ; (iii) the entropy production is positive
definite at the level of the quadratic terms. A more general derivation is due to Banach and
Pierarski (1989) who write an infinite hierarchy of the tensorial symmetric traceless balance
equations, either for classical gases described by the Boltzmann equation, or for quasiparticle
gases (phonons, magnons, rotons, etc) described by the Boltzmann–Peierls equation.
The former method presents the drawback to introduce an infinite number of variables. To
eliminate a maximum number of irrelevant fast variables, we have proposed (Jou and Pavón
1991, Dedeurwaerdere et al 1996) an alternative approach; the basic idea is a renormalization
of the fast variables instead of the more classical procedure which consists of their adiabatic
elimination. A reduction in the number of variables is greatly needed both for practical use and
for comparison with experimental results, which usually involve a limited number of variables.
As an illustration, consider the problem of heat conduction in a rigid solid. The results are
easily transposable to viscous flows, electrical conduction or any other transport phenomenon.
Instead of taking one single non-equilibrium variable, the heat flux q , we introduce higherorder variables, each one being defined as the flux of the preceding one. To be explicit, we
take as variables J (1) , J (2) , . . . , J (n) , where J (n) , a tensor of order n, is the flux of J (n−1) . In
microscopic terms, J (1) = q is a part of the third-order moment of the velocity distribution
function, and J (n) is, correspondingly, a part of its (n+ 2)th-order moment. Up to the nth-order
flux, the Gibbs equation will take the form
ds = T −1 du − α1 v J (1) · dJ (1) − · · · − αn v J (n) ⊗ dJ (n)
(7.27)
Extended irreversible thermodynamics revisited (1988–98)
1083
and the entropy flux will read as
J s = β0 J (1) + β1 J (2) · J (1) + · · · + βn−1 J (n) ⊗ J (n−1)
(7.28)
where all the coefficients αi and βi are generally temperature dependent, and the coefficient
β0 may be identified by comparison with the classical theory as β0 = T −1 . We have limited
ourselves to the simplest form of the entropy and the entropy flux and have not taken into
account contributions of the form J (3) : J (1) J (1) . This approximation is sufficient for the
present purpose. Moreover, since we have in mind linear developments, the temperature is the
local-equilibrium one.
The entropy production is easily derived from (7.27) and (7.28) and is given by
s
σ = −[−∇T −1 + α1 J˙(1) − β1 ∇ · J (2) ] · J (1) · · ·
−
N
X
J (n−1) ⊗ [αn−1 J˙(n−1) − βn−1 ∇ · J (n) − βn−2 ∇ J (n−2) ].
(7.29)
n=3
The above expression suggests the following set of evolution equations for the J (n) :
β 1 τ1
∇ · J (2)
τ1 J˙(1) = −(J (1) + λ∇T ) +
α1
(7.30)
β n τn
βn−1 τn
τn J˙(n) = −J (n) +
∇ · J (n+1) +
∇ J (n−1)
(n = 2, 3, . . . , N)
αn
αn
with µn > 0, as required from the positiveness of the entropy production.
Since J (n) is the flux of J (n−1) , this implies, by the very definition of a flux, that the
coefficient of ∇ · J (n) in the equation for ρ J˙(n−1) must be equal to −1. As a consequence,
ρβn = −αn , a relation which reduces considerably the number of independent parameters.
In the (ω, k) Fourier space, with ω the frequency and k the wavenumber, the hierarchy
of equations (7.30) may be written as a generalized transport law with (ω, k)-dependent
coefficients
J˜(1) (ω, k) = −ikλ(ω, k)T̃ (ω, k)
(7.31)
where J˜(1) (ω, k) and T̃ (ω, k) are the Fourier transforms of J (1) ≡ q (r , t) and T (r , t) while
λ(ω, k) stands for
λ0
(7.32)
λ(ω, k) =
k 2 l12
1 + iωτ1 +
k 2 l22
1 + iωτ2 +
k 2 l32
1 + iωτ3 +
1 + iωτ4
2
2
−1
with ln = βn (µn µn+1 ) > 0. The continued-fraction (7.32) allows us to define a k-dependent
thermal conductivity λ(k) in the steady state, which is similar to earlier results established by
Mori (1965).
After introducing (7.32) into the energy balance equation ρcv Ṫ = −∇ · q , one obtains
the dispersion relation between ω and k:
χk 2
.
(7.33)
−iω =
k 2 l12
1 + iωτ1 +
k 2 l22
1 + iωτ2 +
k 2 l32
1 + iωτ3 +
1 + iωτ4
The value of the phase speed νp = ω/Rek depends on the order of approximation considered
in (7.33) (see Kranys (1989) for the calculation of the phase speed at several orders of
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D Jou et al
approximation in an analogous situation). For ideal gases, the convergence of (7.33) is slow,
and therefore it is necessary to use an asymptotic expression starting from an infinite number
of higher-order fluxes. A proposed scheme is the following. Let us define An (ω, k) by
An (ω, k) = λn (ω, k)/λ0 , where λn (ω, k) is the nth-order approximation to λ(ω, k) in (7.32).
In the asymptotic limit (n → ∞), one has
1
(7.34)
A∞ (ω, k) =
2
1 + iωτ∞ + l∞ k 2 A∞ (ω, k)
with τ∞ and l∞ the limits of τn and ln for n → ∞. This scheme has proved to be sufficiently
accurate for a wide variety of problems in physics. Solving (7.34) with respect to A∞ (ω, k)
results in
p
2
−(1 + iωτ∞ ) ± (1 + iωτ∞ )2 + 4k 2 l∞
.
(7.35)
A∞ (ω, k) =
2 k2
2l∞
The dispersion equation (7.33) thus becomes iω = −χk 2 A∞ (ω, k) which leads in the highfrequency limit to
q
χ
2 k 2 − ω2 τ 2
(7.36)
iω = 2 iωτ∞ ± 4l∞
∞
2l∞
while the corresponding phase speed is
χ
2
=
.
(7.37)
νp∞
2 /χ)
τ∞ − (l∞
By identifying this expression for the phase velocity with the standard expression
νp = (χ/τeff )1/2 , one may introduce a ‘renormalized’ or ‘effective’ relaxation time expressed
by
2
/χ).
τ1eff = τ∞ − (l∞
(7.38)
The interest of this procedure is that it takes account of the presence of all the fluxes although
one single flux, the first-order one, is used in the analysis; the key point is that its relaxation
time is no longer τ1 but rather τ1eff given by relation (7.38).
An analogous development for the viscous pressure tensor would yield the following
expression for the effective relaxation time of the corresponding variable:
02
/ν)
τ2eff = τ2 − (l∞
(7.39)
0
l∞
ν being the kinematic viscosity, and the corresponding correlation length. It was shown by
Hess (1977) that
3kB T 2
3kB T 2
2
02
τ1 ,
τ .
=
l∞
=
(7.40)
l∞
4m
4m 2
By recalling that χ = 5kB T /(3m)τ1 and ν = (kB T /m)τ2 , it is found from (7.38) and (7.39)
11
τ1Grad , τ2eff = 41 τ2Grad where τ1Grad and τ2Grad are the respective relaxation
that τ1eff = 20
times in the 13-moment approximation. These values for the effective times are comparable to
the values τ1exp = 0.40τ1Grad , τ2exp = 0.29τ2Grad obtained by Anile and Pluchino (1984a, b)
who fitted the experimental data for ultrasound velocity.
Clearly the above procedure leads to results which are in better agreement with the
experimental results than the 13-moment approach. It presents the advantage that it does not
ignore a priori all the other fluxes. By considering only the first-order fluxes as independent
variables, one nevertheless obtains a satisfactory qualitative understanding of high-frequency
phenomena. Truly, the above method provides a different way to cut through the hierarchy of
equations: instead of assuming that the higher-order moments vanish, it is supposed that all
of them decay with the same relaxation time as the latter variable included in the description.
Extended irreversible thermodynamics revisited (1988–98)
1085
The hypothesis that all the relaxation times are of the same order of magnitude is not
completely accurate. In fact, according to the Boltzmann equation, they decrease more slowly
for increasing order of the fluxes, which could imply that the maximum propagation speed
in the system may increase without bound. This is not completely surprising, because the
Boltzmann equation is not hyperbolic.
Another field in which the number of moments kept in the description is particularly
relevant is shock waves. One of the original motivations of EIT was to obtain finite speeds
for the propagation of dissipative signals. If this is achieved, the speed of the shock waves
cannot be higher than the highest characteristic speed of the system (Ruggeri 1993). As a
consequence, the structure of shock waves cannot be regular for Mach numbers M exceeding
a critical value Mc (e.g. Boillat and Strumia 1988, Olson and Hiscock 1990a, Khonkin and
Orlov 1993). The problem is that, when limited to the 13-moment approximation, EIT yields
a very low value for Mc , of the order of 1.65, in contrast to experiments, for which a singular
point is not observed. Some authors (Israel 1989) have pointed out this discrepancy as one of
serious problems faced by EIT and, in general, by hyperbolic transport theories.
It is expected that the value of Mc will be raised when higher-order moments are
incorporated in the formalism (Jou and Pavón 1991, Weiss 1995, 1996, Au 1996). However,
this argument is criticized by Ruggeri (1993), who argues that, in principle, each characteristic
speed of the system, and not only the maximum speed, will produce a singularity. However,
Weiss (1995) was able to show that, though it is true that the shock structure should be expected
to become singular at every characteristic speed, numerical calculations with 13, 21 and 51
moments indicate that the structure remains regular for speeds lower than the maximum one.
This opens the possibility to raise Mc by adding higher-order moments: for instance, whereas
for 13 moments Mc = 1.65, for 21 moments Mc = 1.887 and for 51 moments Mc = 2.809.
Whether asymptotic expressions including an infinite number of moments will yield a finite or
an infinite speed of propagation remains an open question, as well as the role of the nonlinear
terms on the value of the critical Mach number (Jou and Pavón 1991, Ruggeri 1993).
A particular example of the hierarchy (7.30) is well suited in the case of phonons and
photons (Larecki and Pierarski 1992, Dreyer and Struchtrup 1993, Blokhin et al 1996). The
energy density and the energy flux are now given as
Z
Z
2
q = c h̄ kf dk
(7.41)
u = ch̄ kf dk;
where k is the wavevector. Then, the higher-order fluxes are expressed by
Z n−1
c
uhi1 ...in i = h̄
khi1 . . . kin i f dk.
(7.42)
k
An analysis based on kinetic theory yields the following hierarchy of equations for the higherorder fluxes:
∂uhi1 ...in−1 i ∂uhi1 ...in li
n
∂uhi1 ...in i
+
= Phi1 ...in i
(7.43)
+
c2
∂t
2n + 1
∂xin
∂xl
with h. . .i denoting the completely symmetrized part of the corresponding expression. In the
relaxation-time approximation, one has for the production terms
1
(7.44)
Phi1 ...in i = − uhi1 . . . in i.
τ
In fact, the analysis of the production terms requires us to take into account the importance
of the various absorption and scattering effects (as, for instance, bremsstrahlung (free–free)
absorption, Thomson scattering with electrons and protons, and so on) at different temperatures,
so that (7.44) is only a suitably simplified expression. Struchtrup (1997) has recently used a
1086
D Jou et al
more general method of moments including powers of the frequency, which allows one not
only to reflect the anisotropy of the non-equilibrium distribution function, but also its spectral
deviation with respect to the Planckian one.
7.4. Comparison with other results
There are other microscopic approaches which are not directly based on the moments of
the distribution function, but which take alternative points of view. For instance, Gorban
and Karlin have proposed interesting modifications of Grad’s method. In a first approach,
Gorban and Karlin (1991) analysed the Chapman–Enskog expansion in terms of the linearized
Grad equation instead of the Boltzmann equation and were able to eliminate the spurious
short-wavelength instability of the classical Burnett approximation, in which sufficiently short
wavelength acoustic waves increase with time instead of decaying. More recently, Gorban
and Karlin (1996a) have shown that not only linear hydrodynamics is stable at all wavelengths
but, in addition, an asymptotic expansion of the acoustic spectrum in the short-wavelength
domain was derived. These results are of interest in EIT, whose main purpose is to describe
hydrodynamics in the short-wavelength regime. Let us also mention that Gorban and Karlin
(1996b) have proposed an alternative to Grad’s method by using the moments of the collision
operator rather than the moments of the distribution function itself as independent variables.
This choice is more sensitive to the details of the interparticle potential, and gives a better
evaluation of the viscosity and the relaxation time for non-Maxwellian molecules. However, as
yet, the thermodynamic consequences of Gorban and Karlin’s model have not been calculated.
Comparison with kinetic theory is not restricted to ideal monatomic gases, but
encompasses more general systems such as molecular gases, and gases with internal degrees
of freedom (Kremer 1992) and mixtures of gases (Uribe and Garcı́a-Colı́n 1993, Khonkin and
Orlov 1994).
Another situation of special interest for the microscopic analysis of EIT is provided by
dilute gases under shear, where many expressions for the microscopic distribution function
have been worked out. Santos and Brey (1991) calculated the velocity distribution of a
dilute gas far from equilibrium for the BGK kinetic equation in a uniform shear flow; the
latter is characterized by a linear profile of the x-component of the local velocity along the
y-axis, a constant density and a constant temperature: this has the advantage that the only
nonzero gradient is ∂νx /∂y = γ̇ . Setting γ̇ij = ∂νi /∂xj , the Boltzmann equation in the BGK
approximation is written as
f − feq
∂f
∂f
=−
+ γ̇ij νj
.
∂t
∂νj
τ
(7.45)
Multiplying this equation by mνi νj and integrating over the velocity space, one finds for the
pressure tensor the following evolution equation:
∂Pij
1
(7.46)
+ (γ̇ik Pj k + γ̇j k Pik ) = − (Pij − pδij ).
∂t
τ
For Maxwell molecules, an exact solution can be derived from which follow some interesting
nonlinear features. For instance, the shear viscosity turns out to be a function of the shear rate
2
2 1
−1
2
cosh
η
sinh
[1
+
9(
γ̇
τ
)
]
(7.47)
η=
0
(γ̇ τ )2
6
with η0 = pτ .
Since the equation for Pij has been found without requiring the explicit form of the velocity
distribution function, it is not surprising to observe that the behaviour of Pij is rather insensitive
Extended irreversible thermodynamics revisited (1988–98)
1087
Table 2. Coefficients of the non-equilibrium contribution to the entropy in the second, fourth and
sixth powers of shear rate (Montanero and Santos 1996).
Theory
S (2)
Boltzmann
−0.5000
Relaxation time −0.5000
MaxEnt
−0.5000
S (4)
S (6)
0.5842 −1.3650
0.2500 −0.0926
0.7500 −1.7593
to the details of the distribution function. Santos √
and Brey (1991) showed that a Chapman–
Enskog expansion√is only convergent for γ̇ τ < 2/3; it becomes divergent for vanishing
velocity at γ̇ τ > 6. A description limited to second-order moment approximation is unable
to capture these features, so that one may ask whether the fourth- and higher-order moments
are necessary.
Montanero and Santos (1996) have expanded the expression of thePnon-equilibrium
entropy up to the sixth order in powers of the shear rate. They use s = n S (n) γ̇ n . Their
results for S (2) , S (4) , S (6) in the case of Maxwell molecules, respectively from the Boltzmann
equation, the relaxation time approximation and the maximum-entropy approach are given in
table 2. It is interesting to note that the values of S (2) coincide in the three cases, in such a way
that the second-order approximation is rather universal, but that the higher-order coefficients
S (4) , S (6) differ. The maximum entropy expression yields the maximum value for S (4) and the
minimum value for S (6) .
It is worth stressing that the non-equilibrium corrections are inversely proportional to some
power of the density and are expected to become increasingly important for decreasing density.
The analysis of rarefied gases should therefore be a domain of special interest for EIT. Since,
in this situation, collisions with the walls become more and more important as compared with
collisions amongst particles, the hypotheses underlying the Boltzmann equation break down
and new kinetic equations must be devised in the dilute gas regime (Zhedanov and Roldughin
1993, Sharipov 1994). However, the thermodynamic aspects of this regime have not yet been
thoroughly studied, in contrast to the form of the transport equations, which have received
much more attention because of their practical interest in astronautics and space sciences.
The full analysis of far-from-equilibrium situations requires us to include nonlinear
contributions; some of them are of thermodynamic origin (namely, those arising from the
generalized equations of state as discussed in section 3), whereas others are of purely dynamical
origin, and their analysis must be carried out from kinetic equations. Recent studies show that
second-order effects are enhanced in ionized gases in the presence of magnetic fields. The
monographs by Eu (1992) and Woods (1993, 1996a, b) contain a wide bibliography on these
topics.
7.5. Non-equilibrium entropy, the H -theorem and the second law
In the earlier formulations of EIT it was assumed that the macroscopic entropy was strongly
related to the microscopic Boltzmann entropy: the difficulties concerning the choice of relevant
variables or the positive character of the entropy production have made it apparent that the
problem is more challenging than previously thought, because one is not working with the exact
solution of the Boltzmann equation but with approximate distribution functions. Therefore, a
comparison between the macroscopic entropy of EIT and the Boltzmann definition of entropy in
kinetic theory of gases, or the Gibbs definition of entropy in statistical physics and information
theory, may shed some light towards the definition of a non-equilibrium entropy (Chen and Eu
1993, Eu 1995a, b, Ichiyanagi 1995a, b, 1997).
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D Jou et al
A second point is that the thermodynamic entropy is a function of macroscopic variables.
Therefore, assuming the existence of a macroscopic entropy implies that one selects among
the several possible microscopic distribution functions those depending parametrically on
quantities which have a clear macroscopic meaning or, at least, which are macroscopically
measurable and controllable. In principle, one can construct a wide range of microscopic
distribution functions but their physical content is not necessarily clear.
Eu (1992, 1995c, 1997) has paid special attention to this point. In contrast with
Grad’s assumption (7.21), where the distribution function is expanded in terms of Hermite
polynomials, Eu proposed a modified moment method (Eu 1992) in which the logarithm of the
distribution function rather than the distribution function itself is expanded in terms of such
polynomials. More explicitly, he defines a canonical distribution function as
X
Yk H (k) (c) − µ
(7.48)
f c = exp − β H (c) +
k>1
where H (c) is the Hamiltonian, H (k) refers to the moment of order k of the distribution
function, namely the kth order tensor of the Hermite polynomial of the particle velocity, and
Yk designates the corresponding conjugate variable and µ is the chemical potential. Clearly,
Eu makes a distinction between the Boltzmann entropy, defined by (7.3), or in another notation
s = −kB hf (ln f − 1)i
(7.49)
and the so-called calortropy, defined as (Eu 1995b, 1997)
ψ = −kB hf (ln f c − 1)i.
(7.50)
According to Eu, it is the calortropy (7.50) rather than the Boltzmann entropy (7.49) which
provides the microscopic basis for the phenomenological entropy used in EIT (note that Eu
reserves the name entropy for the Boltzmann entropy, and applies the term calortropy to both
the microscopic quantity (7.50) and its macroscopic counterpart).
The difference between the Boltzmann entropy and the calortropy is called relative entropy
by Eu, and is defined as
s[f |f c ] ≡ ψ − s = kB hf (ln f − ln f c )i.
(7.51)
By Klein’s inequality, it is shown that the relative entropy (7.51) is always positive, which
implies that s 6 ψ. The relative entropy is related to the loss of information when the
description of the many-particle system is projected onto a thermodynamic space. The relative
entropy may thus be considered as a hybrid quantity resulting from the mixing of a kinetic and
a thermodynamic theory.
By using the concept of relative entropy, Ichiyanagi (1997) was able to clarify the relations
between the second law, i.e. the increase of macroscopic entropy in isolated systems, and the
H -theorem, reflecting the irreversibility of the microscopic Boltzmann equation. If f is an
exact solution of the Boltzmann equation (or of the Fokker–Planck equation or more general
kinetic equations) then ∂s/∂t as defined in (7.49) is necessarily positive (or zero) according
to the H -theorem. However, this is no longer true if f is an approximate solution of the
Boltzmann equation. Since Grad’s expansion is only approximate, the corresponding entropy
production may be negative in some situations. Therefore, the positiveness of σ s could serve
to select which approximate solutions fapprox are admissible.
8. Stochastic processes
Stochastic processes have been used as a microscopic basis for the derivation of generalized
transport equations with memory. In particular, the telegrapher equation is a typical model
Extended irreversible thermodynamics revisited (1988–98)
1089
equation for the description of some classes of stochastic processes. Solutions of the telegrapher
equation have been obtained by many authors (see Masoliver and Weiss (1994, 1996),
Masoliver et al (1993) and Godoy and Braun (1994) for a wide bibliography), and its relation
with dichotomous noise has also been examined (Masoliver 1993). A microscopic derivation
of the telegrapher equation for non-Markovian stochastic processes has also been calculated
(Olivares-Robles and Garcı́a-Colı́n 1994, 1996).
8.1. Persistent random walk
The telegrapher equation for the diffusion of particles can be derived from several microscopic
grounds, for instance by assuming a persistent random walk (i.e. a random walk in which
the jump directions in any two consecutive intervals are correlated) or from a dichotomous
model in which particles switch between two different states, from a random walk with a
continuous distribution function of pausing times, or from differential stochastic equations
with telegrapher noise. The persistent random walk was introduced by Fürth (1920) to model
diffusion in biological problems and was developed by Taylor (1921) to analyse turbulent
diffusion; later it was popularized by the classical work of Goldstein (1951) on diffusion by
discontinuous movements, and since then it has been widely studied. It provides a good model
for the description of heat transport, diffusion of light in turbid media or absorption of laser
radiation in biological tissues, because of its ability to describe the effects of forward scattering.
Assume that at the initial time t = 0, all the particles occupy the position x = 0. At
time t = t0 , half of the particles jump a distance d to the right and half to the left. Now,
after each interval t0 , every particle jumps a distance d, with a probability p of jumping in the
same direction as the previous jump and a probability q = 1 − p of jumping in the opposite
direction. In a normal random walk, the probability of jumping to the right or to the left is
the same, i.e. p = 21 , so that there is no correlation between the speed direction at successive
jumps. It is easily shown (e.g. Jou et al 1996a) that such a correlation leads to the telegrapher
equation
∂ 2 w ∂w
∂ 2w
= τ ν2 2
+
(8.1)
2
∂t
∂t
∂x
where w(t, x) is the probability density of the distribution of particles, τ the relaxation
time τ and τ ν 2 the diffusion coefficient. There is no drift term because it is assumed that
there is no external force. The above result concerns one-dimensional motion. At higher
dimensions one does not recover the telegrapher equation, but a more general expression
involving spatial derivatives of order higher than two (Masoliver et al 1993, Garcı́a-Colı́n and
Olivares-Robles 1995, Olivares-Robles and Garcı́a-Colı́n 1994, 1996), so that one is led to a
non-local generalization of Fick’s law.
Generalizing to higher orders one obtains, instead of (8.1), an expression of the form
X ∂ i w X ∂ 2j w
ai i =
bj 2j
(8.2)
∂t
∂x
i
j
τ
where the explicit expressions for the coefficients ai and bj can be directly calculated (Jou et al
1996a). As shown by Rosenau (1993), the telegrapher equation (8.1) satisfactorily reproduces
the spectrum of the original discrete process of the correlated random walk for all wavelengths.
This is remarkable because, in general, the short-wavelength limit of a continuum model differs
widely from the corresponding discrete microscopic model. Indeed, the well known Fokker–
Planck equation, which provides a good description of completely uncorrelated random walks
at long times, does not reproduce microscopic results in the short-wavelength regime. The
reason for the much more satisfactory behaviour of the telegrapher’s equation is that the latter
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D Jou et al
preserves the characteristic speed of the walker, d/t0 , in contrast with the Fokker–Planck
equation.
The telegrapher equation may not only be derived for discontinuous, but also for
continuous processes in one dimension, whereas more general equations must be used in
two or more dimensions (Olivares-Robles and Garcı́a-Colı́n 1994, 1996). A simple model
for the analysis of the persistent random walk is the so-called two-layer model (Camacho and
Zakari 1994); in its most intuitive form, it consists of a system of particles which jump at
random times between two states, 1 and 2, each of them with a velocity (ν1 , ν2 , respectively)
along the x-axis. Denoting by Pi (x, t)(i = 1, 2) the probability density of finding a particle
in state i at position x at time t, the evolution equations for Pi have the form
∂P1
∂P1
+ν
= −m
∂t
∂x
(8.3)
∂P2
∂P2
−ν
=m
∂t
∂x
where m is the rate of particle exchange between both states and where we have assumed
ν1 = −ν2 = ν. By taking m = r(P1 − P2 ), with r a constant, one obtains a relaxational
Maxwell–Cattaneo equation for the particle flux J = P1 ν1 + P2 ν2 . The entropy is
kB
J
1
J
kB
J
1
J
S(P , J ) = −
P+
ln
P+
−
P−
ln
P−
(8.4)
2
ν
2
ν
2
ν
2
ν
where P = P1 + P2 is the particle concentration. This expression reduces to the equilibrium
one when J = 0 and is completely nonlinear in J ; therefore it extends the non-equilibrium
entropy beyond the second order in J .
8.2. H -theorem for telegrapher-type equations
It may be asked whether a H -theorem can be formulated for kinetic equations of the telegrapher
type. Consider, for instance, the following equation
∂ 2 f ∂f
= ∇ · D0 [kB T ∇f + (∇u)f ]
+
(8.5)
∂t 2
∂t
which generalizes the Smoluchowski equation describing the dispersion of non-interacting
Brownian particles in an external potential density u(x); f (x, t) is the probability distribution
function of the particles, D0 the inverse of the friction coefficient and τ a relaxation time. By
analogy with kinetic theory, the entropy S is defined by
Z
S = −kB f (x, t)[ln f (x, t) − 1] dx.
(8.6)
τ
Since one has in mind a system atRconstant temperature T , it is more convenient to use a ‘free
energy’ F = U − T S, with U = dx f (x, t)u(x). The evolution of F is thus described by
Z
∂f
dF
=
(kB T ln f + u(x)) dx.
(8.7)
dt
∂t
Substituting in (8.7) the expression for ∂f/∂t drawn from (8.5), one obtains
Z
Z
∂ 2f
1
dF
= −τ dx 2 [kB T ln f + u(x)] − D0 dx [kB T ∇f + (∇u(x))f ]2
(8.8)
dt
∂t
f
as is easily seen by integration by parts and assuming that the flux vanishes at the boundaries.
The second term in the right-hand side of (8.8) is negative, because the friction coefficient
is positive, but the first term has no definite sign. When the relaxation time τ is zero, (8.5)
Extended irreversible thermodynamics revisited (1988–98)
1091
reduces to the Smoluchowski equation and dF /dt is definite negative. However, dF /dt may
be non-negative when τ is different from zero.
To circumvent this difficulty let us introduce, in analogy with EIT, a generalized entropy S
which depends not only on f but also on the probability flux j , the latter being defined through
the conservation equation
∂f
(8.9)
+ ∇ · j = 0.
∂t
The generalized entropy S is defined as (Camacho and Jou 1992)
Z
(8.10)
S = −kB dx [f (ln f − 1) + α(f )j 2 ],
where α(f ) is an undefined coefficient depending on f , to be identified below. The evolution
equation for the generalized free energy F = U − T S obtained by using this generalized
entropy (8.10) is given by
Z
dF
= dx {(∂f/∂t)[u(x) + kB T ln f ] + kB T α 0 j 2 (∂f/∂t) + 2kB T α j · (∂ j /∂t)}
(8.11)
dt
where α 0 stands for dα(f )/df . After integration by parts one obtains
Z
dF
= dx {j · [∇(u(x) + kB T ln f + kB T α 0 j 2 ) + 2kB T α(∂ j /∂t)]}. (8.12)
dt
To preserve the negative character of dF /dt, it is observed that j cannot be independent of the
term inside the brackets; the simplest relation for j is then
j = −D[∇(u(x) + kB T ln f + kB T α 0 j 2 ) + 2kB T α(∂ j /∂t)].
(8.13)
Introducing (8.13) into (8.12) it is found that dF /dt < 0. Now assuming that the term in j 2 is
negligible, one recovers (8.5) provided the coefficient α is identified as α = τ (2kB T D0 f )−1 .
This result is important as it shows that it is possible to formulate an H -theorem for the kinetic
telegrapher equation at the condition to modify the Boltzmann definition of entropy.
A formulation of the H -theorem for a generalized master equation involving relaxational
terms (Hongler and Streit 1990) was also proposed by Vlad and Ross (1993). A more general
approach was given by Grmela and Jou (1993) by means of an extended version of the kinetic
theory of gases in which not only the one-particle distribution function f1 (r , c, t) but also the
two-particle distribution function f2 (r1 , r2 , c1 , c2 , t) is an independent variable. In this study
f2 plays the role of the fast variable, which tends in a short time to the usual value f2 ≈ f1 f1
adopted a priori in the Boltzmann equation.
9. Computer simulations
Computer simulations have fostered much progress in nonlinear statistical mechanics and
provide a tool to study systems barely accessible to direct experimental observations; they
are also useful to assert, at least in some specific situations, the foundations of macroscopic
formalisms. In particular, computer simulations are helpful in the formulation of a nonequilibrium thermodynamics beyond the local-equilibrium approximation. On the other hand,
computer simulations must rely on some given assumptions about the meaning of temperature
(which is usually identified as the kinetic temperature) and about the kind of average to be
performed; as a consequence, a comparison with macroscopic formalisms may shed a critical
view on these crucial and subtle matters. Non-equilibrium equations of state have been studied
widely by means of computer simulations by Evans and collaborators (e.g. Evans and Morriss
1990).
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D Jou et al
These authors carried out their calculations for a system of particles subject to a LennardJones interaction potential for several fixed values of the density and the energy, when a constant
shear rate γ̇ is acting on the system and when a steady state is maintained. Moreover, a suitable
thermostating procedure is introduced which removes the dissipated heat to keep the energy
(kinetic temperature) constant.
Evans and Morriss (1990) give the microscopic equations describing the motion of the
molecules under the action of a velocity gradient ∇ν as
pi
ṙi =
+ (∇ ν )T · ri
(9.1)
m
ṗi = Fi − (∇ ν )T · pi − α pi
where Fi is the external force acting the molecule i of mass mi and α pi (with pi the momentum
of the molecule i) is a Gaussian thermostat which removes energy from the system so as to
keep the internal energy constant. This is achieved by imposing that
V P ν : (∇ν)
(9.2)
α = −P 2
i (pi /mi )
where V is the total volume. This description is not fully realistic, as it implies that heat
is removed at the same point where dissipation is produced, whereas in real situations it is
eliminated across the boundaries of the system. Furthermore, the Gaussian thermostat produces
sharply defined kinetic energy, whereas potential energy is distributed canonically. A better
choice for the thermostat is the so-called Nosé–Hoover thermostat, which yields results in
agreement with the canonical ensemble, not only for the potential but also for the kinetic energy.
Since the rate of viscous heating is quadratic in the shear rate, both thermostats obviously
lead to the same results in the linear regime. The aim of these simultations is to obtain the
transport coefficients of the gas, with a greater precision than the usual Green–Kubo techniques.
Although most practioners of simulations do not pay attention to the thermodynamic potentials,
Evans and co-workers have devoted some effort to determine the entropy or the free energy.
Their numerical results are thus useful to check recent ideas on non-equilibrium
temperature and pressure, and also on fluctuations in non-equilibrium states (Baranyai and
Cummings 1995a, b). In their earlier calculations, Evans did not calculate the entropy
and defined simply the temperature in terms of the average kinetic energy of the particles.
Later, Evans (1989) computed the entropy and the temperature for an isoenergetic planar
Couette flow at low densities of soft sphere particles with an interaction potential of the form
φ(r) = ε(r/σ )−12 . The large mean free paths in this low-density regime require very long runs
to achieve an accuracy comparable to that for dense fluids; as a matter of fact, the results were
achieved after 15 million time-step calculation runs. Table 3, which shows some of Evans’
results, makes evident the difference between the kinetic (local-equilibrium) temperature T and
the (non-equilibrium) thermodynamic temperature θ defined as the derivative of the internal
energy with respect to the entropy at constant volume and mole number. It is to be noted that
θ < T in agreement with the results of EIT (see section 3). In table 3, the quantities ρ, γ̇ and
s are given in terms of nm, (ε/mσ 2 )1/2 and nkB , respectively; T and θ are expressed in units
of ε/kB .
Furthermore, Evans has calculated the non-equilibrium pressure, defined as π =
−(∂U/∂V )γ̇ , and has compared it with the kinetic pressure p identified as one-third of the
trace of the pressure tensor. Some of his results are reproduced in table 4. It is evident from
these results that the differences between the values of θ and T and between those of π and
p are not negligible in the presence of high shear rates. Evans (1989) has noticed that the
numerical data for the thermodynamic pressure π agree with the minimum eigenvalue of the
pressure tensor. We have found in (3.23) a similar result for a gas submitted to a heat flux.
Extended irreversible thermodynamics revisited (1988–98)
1093
Table 3. Values for the non-equilibrium entropy s, the kinetic temperature T and the thermodynamic
temperature θ at energy u = 2.134 for different densities and shear rates γ̇ . All the quantities are
expressed in units of the parameters of the molecular potential ε, σ , and kB (Evans and Morriss
1990).
ρ
γ̇
s
T
θ
0.100
0.100
0.075
0.075
0.5
1.0
0.5
1.0
5.653
5.392
5.852
5.499
2.171
2.169
2.190
2.188
2.048
1.963
2.088
1.902
Table 4. Values of the local-equilibrium pressure p versus the non-equilibrium pressure π at energy
u = 2.134 and density ρ = 0.100 for different values of the shear rate γ̇ (Evans and Morriss 1990).
γ̇
p
π
0.5
1.0
0.245
0.247
0.145
0.085
The main difference between the results of these numerical simulations and EIT is that the
former predict a non-analytic dependence of p and E on the shear rate γ̇ while EIT exhibits
a dependence in γ̇ 2 . Indeed, it is found from numerical simulations that the free energy
F (T , V , γ̇ ) is of the form
F (T , V , γ̇ ) = Feq (T , V ) + F1 (T , V )γ̇ 3/2 .
(9.3)
A strict comparison with EIT requires a theory valid at very high values of the shear rate.
Indeed, computer simulations have been performed at high values of γ̇ in order to emphasize
the effects of shear. In contrast, the quadratic approximation supposes that γ̇ remains small.
Nevertheless, the transition from a γ̇ 2 to a γ̇ 3/2 regime may be apprehended qualitatively in
the context of EIT (Bidar 1997). To achieve this goal, let us start from the following EIT
expression of the free energy:
F (T , V , γ̇ ) = Feq (T , V ) + 21 τ V ηγ̇ 2 .
(9.4)
Let us now introduce in this expression the following value of η(γ̇ ) obtained by fitting the
results (6.31) of information theory (Bidar et al 1996, Bidar 1997):
η0
η=
(9.5)
[1 + (aτ γ̇ )n ]1/n
where η0 is the value of the shear viscosity in the low shear-rate limit, and a and n are
fitting parameters. It must be noted that (9.5) does not follow from first principles; it is,
rather, an heuristic simplification of the more involved implicit expression (6.31) derived from
information theory, but it captures the essential features of the asymptotic behaviour of the
pressure tensor at high and low values of the shear rate. Substitution of (9.5) in (9.4) yields
η0
τV
F (T , V , γ̇ ) = Feq (T , V ) +
γ̇ 2 .
(9.6)
2 [1 + (aτ γ̇ )n ]1/n
Note that the asymptotic behaviour of F at low γ̇ is of the form γ̇ 2 , whereas for high γ̇ it
behaves like γ̇ ; therefore, at the intermediate regime F is indeed expected to depend on γ̇ as
γ̇ 3/2 .
Evans and co-workers have also studied the behaviour of a fluid in a non-homogeneous
shear flow and in a Poiseuille flow. They observed that a heat flow is produced even for a
uniform distribution of kinetic temperature. This means that such a temperature is no longer
1094
D Jou et al
related to the zeroth principle of thermodynamics in non-equilibrium situations. Their results
are compatible with a generalized Fourier equation of the form (Todd and Evans 1997)
q = −λ∇θ + β∇ · P ν + β 0 ∇ · (P ν · P ν )
(9.7)
wherein terms proportional to ∇ γ̇ arise both from the non-classical part of ∇θ and ∇ ·(P ·P ν ).
The contribution of the last term is of the same order as that of the non-equilibrium temperature;
this makes it difficult to ascertain which proportion of the effects of ∇ γ̇ 2 is due to ∇θ and
which is due to ∇ · (P ν ·P ν ), and therefore, for the moment, it has not given definitive evidence
of the non-equilibrium temperature.
2
ν
C. Applications
Having reviewed and discussed the foundations of EIT, it may be asked which kind of problems
will specifically be solved by using the methods and results of EIT. A first series of applications
was presented by Nettleton and Sobolev (1995, 1996). Here, we emphasize some aspects not
considered in their review to which the interested reader is referred for some complementary
details. Among the applications, we have selected several subjects of special practical interest,
such as non-classical heat transport, polymer physics, non-Fickian diffusion, transport in
submicronic devices, and dielectric relaxation, as well as some other topics like superfluids,
nuclear collisions, and cosmological models, which are appealing from a more theoretical
point of view.
10. Non-classical heat transport
It was shown that Cattaneo’s model is particularly well suited for analysing short-time or
high-frequency heat transport beyond Fourier’s law. These situations are found, for instance,
in explosions, the heating of metals by short laser pulses, or the fast compression of solid
hydrogen pellets by means of laser pulses to achieve nuclear fusion. However, the Cattaneo
equation (2.4) remains silent about the spatial microstructure, which may play a role at the
scale of very short wavelengths, in miniaturized devices, or in materials with microstructure
(Sobolev 1994, Tzou 1997) whose transport properties depend, in general, on the ratio of the
mean free path of the quasiparticles (e.g. phonons) and the characteristic length describing the
microstructure as, for instance, the width of the layers in a layered structure. A well known
model including some non-local effects is that of Guyer and Krumhansl (1964) which gives a
satisfactory description of heat pulse propagation in dielectric crystals at very low temperature.
However, even this model presents some limitations: it is a linearized equation; further, when
coupled to the classical energy equation, it predicts infinite speed of propagation at very large
frequencies; and therefore it is unable to describe ballistic propagation. An important question
is then to ask how far EIT is able to describe not only high-frequency but also short-wavelength
phenomena.
10.1. A generalized Guyer–Krumhansl model
These observations have motivated the formulation of an extension of Guyer–Krumhansl’s
result incorporating nonlinearities and spatial inhomogeneities (Lebon et al 1994b, 1998,
Valenti et al 1997). This has been achieved by complementing the pair of variables T (or u)
and qi by an additional one, say Qij related to the flux of the heat flux. The new variable Qij ,
a tensor of rank two, is, however, of different nature to T and qi as it does not appear in the
usual balance equations; moreover, Qij is not directly experimentally measurable. For this
Extended irreversible thermodynamics revisited (1988–98)
1095
reason, Qij may be referred to as an internal variable because it is not controllable from the
external world. From the kinetic point of view, Qij is the fourth moment of the non-equilibrium
distribution with respect to velocity and given by
Z
(10.1)
Qij = mC 2 Ci Cj f dc.
Therefore, it is justified to assume that Qij is symmetric (Qij = Qj i ).
By analogy with the viscous pressure tensor in hydrodynamics, we may decompose Qij
0
into a bulk and a deviatoric part according to Qij = Qδij + Qij where Q = 13 Qkk . The set of
0
independent variables is now formed by T (or u), qi , Q and Qij whose behaviour is described
at the lower order of approximation by the following evolution equations (Lebon et al 1994,
1998, Valenti et al 1997):
τ1
0
∂qi
= βλT 2 Qij,j + β 0 λT 2 Q,i − qi − λT,i
∂t
(10.2)
0
0
∂ Qij
sym
τ2
= −Qij − ηqhi,j i
∂t
∂Q
= −Q − ζ qi,i
τ3
∂t
(10.3)
(10.4)
0
sym
where τ1 , τ2 , τ3 stand for the relaxation times of qi , Qij and Q, respectively, qhi,j i is the
symmetric part of the deviator of qi,j , and β and β 0 are parameters which describe non-local
effects and which turn out to be related to the non-classical contribution to the entropy flux
as seen in (10.8); η and ζ are supplementary phenomenological coefficients which play an
analogous role as the shear viscosity and the bulk viscosity in (2.34)–(2.36). It is instructive
0
to examine the particular case where τ2 = τ3 = 0. After substitution of Qij and Q derived
from (10.3) and (10.4) in (10.2), one obtains
∂qi
+ λT,i + qi = −λT 2 [ 21 βηqi,jj + ( 16 βη + β 0 ζ )qj,j i ]
(10.5)
τ1
∂t
when it is assumed that η and ζ are constant. Comparison of (10.5) with the Guyer–Krumhansl
relation
∂qi
+ λT,i + qi = 15 cs2 τN τR (qi,jj + 2qj,j i )
(10.6)
τR
∂t
leads to the following identifications:
τ1 = τR ,
1
λβηT 2
2
= − 15 τR τN cs2 ,
λβ 0 ζ T 2 = − 13 τR τN cs2
(10.7)
where τN is the characteristic time of normal processes, i.e. phonon–phonon interactions that
conserve momentum; τR is the relaxation time of the dissipative processes due to phonon–
phonon collisions which do not preserve momentum and cs is the Debye phonon velocity.
The non-local effects of the right-hand side modify the profile of heat pulses during its
movement. This is not true with Cattaneo’s equation, according to which a heat pulse of a
rectangular shape will keep its rectangular form, but increasingly shrunken in height, while it
is moving; in contrast, it will acquire a slightly spread bell shape on the basis of the Guyer–
Krumhansl equation, in agreement with experimental observations (Tzou 1997).
The model, whose basic relations are (10.2)–(10.4), involves eight parameters, namely λ,
β, β 0 , η, ζ , τ1 , τ2 , τ3 which are, generally, temperature dependent. Interesting information
about the sign of these coefficients and relations between them is provided by the second law
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D Jou et al
of thermodynamics, stating that the entropy production σ s is a positive-definite quantity. From
the usual procedure whose details can be found in Lebon et al (1994b) and Valenti et al (1997)
and supposing that the entropy flux is given by
0
Jis = T −1 qi + β Qij qj + β 0 Qqi
(10.8)
s
it is easily checked that positiveness of σ given by
0
0
1
β
β0
qi qi − T 2 Qij Qij − T 2 Q2
(10.9)
T σs =
λT
η
ζ
imposes the restrictions λ > 0; β/η < 0; β 0 /ζ < 0.
The corresponding Gibbs equation takes the form
0
0
β
β0
τ1
(10.10)
qi dqi − T τ2 Qij dQij − T τ3 Q dQ
df = −s dT +
λT
η
ζ
where f is the specific Helmholtz free energy. Expanding f around local equilibrium and
imposing that f is minimum at (local) equilibrium, it is found that τ1 > 0, τ2 > 0 and τ3 > 0.
Note that, if a time and space Fourier transform of (10.7) is used, one is led to a frequency- and
wavelength-dependent effective thermal conductivity, which is a truncated version of (7.32)
(with τ2 = 0 and l2 = 0). Some authors (Jäckle 1990, Jäckle and Pieroth 1990) have assumed
that not only the thermal conductivity but also the specific heat depend on the frequency: this
kind of model is useful in undercooled or very viscous fluids, but we do not deal with them
here.
In parallel with the above considerations, the hydrodynamic phonon flow, whose
microscopic analyses may be found in Larecki and Pierarski (1992) and Guyer (1994), is also
worthy of attention. Poiseuille flow of phonons may be observed in cylindrical heat conductors
of radius R when the mean free paths `N = c0 τN and `R = c0 τR satisfy `N `R R 2 and
`N R. In this case, (10.7) reduces in the steady state (∂qi /∂t = 0, qi,j = 0) and for qi = 0
to
3 τN
qi,jj = 0
(10.11)
T,i −
5 ρcv
where use is made of the well known result λ = 13 ρcv cs2 τR . This form is similar to the
equation describing Poiseuille flow of Newtonian fluids provided that qi , T,i , and 35 (τN /ρcv )
are regarded as the velocity νi , the pressure gradient p,i and the viscosity η, respectively.
In contrast to the heat conduction regime, where the total rate of heat transfer across the
cylinder for a given temperature gradient is proportional to R 2 , in the Poiseuille phonon flow
it is found to be proportional to R 4 . This difference allows one to distinguish which regime
is actually occurring. The apparent viscosity in the Poiseuille flow regime provides a way to
determine the normal collision times τN , while the resistive collision time τR is usually obtained
by measuring the thermal conductivity. Let us mention that Jou and Casas-Vázquez (1990)
investigated the effects resulting from the introduction of the non-equilibrium temperature θ
in phonon hydrodynamics, which imply a dependence of the effective conductivity on the
average value of the heat flux.
10.2. Onsager relations
In classical irreversible thermodynamics, the evolution equations of the state variables a α can
be given the general form
X
∂f
∂a α
Lαβ β
(10.12)
=−
∂t
∂a
β
Extended irreversible thermodynamics revisited (1988–98)
1097
where ∂a α /∂t is the thermodynamic flux, f is the free energy, ∂f/∂a β the thermodynamic
force, and Lαβ the phenomenological coefficients. The latter obey the well known Onsager–
Casimir relations
Lαβ = ±Lβα
(10.13)
wherein the sign + (respectively −) refers to state variables a α and a β with the same
(respectively different) parity under time reversal.
It may be asked whether the Onsager–Casimir relations remain satisfied when the classical
evolution equations (10.12) are replaced by more complicated equations like (10.2)–(10.4).
Making use of (10.10) and assuming that the coefficients in (10.2)–(10.4) are constant, we can
reformulate the evolution equations (10.2)–(10.4) as


λ
ληT ∂  ∂f  λζ T ∂
λT ∂f
∂f
∂qi
− T,i (10.14)
=−
− 2
−
0
∂t
τ1 τ2 ∂xj
τ1 τ3 ∂xi ∂Q
τ1
τ1 ∂qi
∂ Qij


∂Qhij i
η  ∂f  ληT ∂
∂f
=
−
(10.15)
0
∂t
βT
τ1 τ2 ∂xhj ∂qii
∂ Qij
∂f
∂Q
λζ T ∂
∂f
ρ
=−
.
(10.16)
+ 0 2
∂t
τ1 τ3 ∂xk ∂qk
β τ3 T ∂Q
These equations may formally be cast in the form
X
X
∂f
∂f
∂a α
=−
Lαβ β −
M αβ ∇ ·
+ ···
∂t
∂a
∂a β
β
β
(10.17)
where a α stands for qi , Qhij i and Q, respectively. It is directly checked from (10.14)–(10.16)
that
ληT
M qQ = M Qq =
τ1 τ2
(10.18)
λζ
T
M qQ = M Qq =
τ1 τ3
or more generally M αβ = M βα . These reciprocity relations, which are derived on purely
macroscopic grounds, may be considered as generalizations of the Onsager–Casimir relations.
It is worth noticing that the argument leading to the symmetry of the coefficients M αβ
parallels the demonstration given by Onsager himself, who postulated that the fluxes are the
time derivatives of the state variables while the forces are the derivatives of a thermodynamic
potential with respect to the state variables. These precepts are observed in (10.14)–(10.16)
wherein the ‘fluxes’ are the time derivatives of the basic variables qi , Qij and Q, while the
forces are the gradients of the derivatives of f with respect to qi , Qij and Q, respectively. It is
rather natural that the gradients appear in our analysis as we are dealing with a non-local theory.
There is, however, one important difference to the Onsager–Casimir results: it concerns the
change of sign. Indeed, the quantities qi and Qij have opposite parities under time reversal as
Qij arises from a higher-order moment with respect to molecular velocity that qi . Therefore,
one should expect skew symmetry according to the Onsager–Casimir rules. But instead it
is seen that relations (10.18) exhibit symmetry properties. The above observations seem to
indicate that, under time reversal, microscopic reversibility requires not only that the sign of
the time be reversed but also that of operator nabla: ∇ → −∇. This result was justified among
others by Ferrer (1991) and Vasconcellos et al (1994) on the basis of microscopic arguments.
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D Jou et al
10.3. Heat propagation velocity
Using the relation U = −T 2 [∂(F /T )/∂T ] and the integrated form of (10.10), one is led to
(Lebon et al 1994b, Valenti et al 1997)
1 2 d τ1 1 2 d τ2 0 0
1 2 0 d τ3
Qij Qij + T β
(10.19)
Q2
u = ueq − T
qi qi + T β
2 dT λT 2
2
dT η
2
dT ζ
where ueq is internal energy at equilibrium, so that dueq /dT = ceq > 0 where ceq denotes the
heat capacity at equilibrium.
At this stage of the analysis, it is convenient to introduce some simplifying hypotheses.
If it is assumed that the quantities τ2 , η, τ3 and ζ appearing in the evolution equations (10.3)
0
and (10.4) for Qij and Q do not vary sensibly with temperature, u will only depend on T and
qi and, according to (10.19), one has
u = ueq + a(T )qi qi
(10.20)
with
1
d τ1 a(T ) = − T 2
.
(10.21)
2 dT λT 2
We now determine the speed of propagation of heat waves (also referred to as second sound)
in a rigid crystal on the basis of the model described by (10.2)–(10.4). Consider a smooth
0
surface 6 where the quantities T , qi , Qij and Q are continuous, but discontinuities between
their first derivatives are permitted. Let us denote by ν the normal wave speed, the unit
normal vector to 6 by ni , and the jump of the first derivatives across 6 by δ. On making
the standard transformation ∂t → −νδ; ∂xi → ni δ, the set of the energy balance equation
and equations (10.2)–(10.4) provides the following homogeneous algebraic system for the
discontinuities, after use is made of (10.20):
νuT δT + 2νaqi δqi − ni δqi = 0
(10.22)
0
ντ1 δqi = −βλT 2 nj δ Qij − β 0 λT 2 ni δQ + λni δT
(10.23)
0
ντ2 δ Qij = η[ 21 (nj δqi + ni δqj ) − 13 nk δqk δij ]
ντ3 δQ = ζ ni δqi .
(10.24)
(10.25)
This set has non-trivial solutions on condition that the following characteristic polynomial
is satisfied:
λ
λ
P (ν) ≡ uT ν 2 + 2 aqn ν − (1 + γ T 2 uT ) = 0
(10.26)
τ1
τ1
where uT is the derivative of internal energy u with respect temperature T , and qn stands for
qi ni while γ is a positive constant given by
2 βη β 0 ζ
γ =−
+
> 0.
(10.27)
3 τ2
τ3
Equation (10.26) admits real solutions if and only if
2
λ
λ
aqn + (1 + γ T 2 uT )uT > 0
τ1
τ1
da 2
q > 0. When da/dT < 0, there exists an upper
i.e. for uT > 0, or from (10.20), ceq + dT
bound on |q| given by
s
−1
da
.
(10.28)
|q| < qcrit = −ceq
dT
Extended irreversible thermodynamics revisited (1988–98)
1099
0
At equilibrium, for which T is uniform and equal to Teq and where, in addition, qi = Qij =
Q = 0, it follows from (10.26) that the velocity of the propagation is simply given by
2
=
νeq
λ
(1 + γ T 2 ceq ).
τ1 ceq
(10.29)
By using the identification given by (10.6), it is possible to express (10.29) in terms of the
relaxation times τR and τN of the resistive and normal phonon–phonon collisions; it is found
that
λ
1
1
4
2
=
+ τN
+
(10.30)
c2 .
νeq
τR ceq 3
5τ2 τ3 2
In comparison with Coleman and Newman’s model (1988), equation (10.29) contains an extra
contribution proportional to γ ; the latter is given by expression (10.27), and in the present
model it is a constant quantity related to the various coefficients appearing in the evolution
equations for the fluxes; the value of γ can be determined from experimental measurements
(Torrisi and Valenti 1992, Valenti et al 1997).
It is worth stressing that the extra term γ T 2 ceq in expression (10.29) is certainly not
negligible, as it is of the order of 0.3 for both NaF and Bi, in the temperature range of interest.
Within the limit τ2 → 0, τ3 → 0 which corresponds to the range of application of the Guyer–
2
Krumhansl relation, νeq
tends to infinity, as stated above.
At temperatures sufficiently low to make the frequency of the resistive collisions very
small, τR−1 → 0, the wave propagation remains finite, and is given by
1
1
4
2
= τN
+
(10.31)
c2 .
νeq
3
5τ2 τ3 s
In the limit of a high frequency of momentum-conserving phonon–phonon collisions, τN−1 → 0
2
=
which corresponds to the range of validity of Cattaneo’s equation, (10.30) simplifies to νeq
1
2
λ/τR ceq . Since for R-processes, the heat conductivity λ is related to τR by λ = 3 ρcv cs τR , one
2
recovers the well known expression for the second-sound velocity νeq
= 13 cs2 .
10.4. Comparison with experimental results
Experiments on second-sound propagation in high-purity crystals at low temperature have
been performed by perturbing systems at uniform temperature. In that respect, to compare our
theoretical results with experimental observations, we need only the simplified mathematical
expressions derived in the particular case of equilibrium. Nevertheless, these results provide
interesting information about the values of the coefficients a(T ) and γ . Measurements have
been carried out on NaF and Bi samples by Jackson and Walker (1971) and Narayamurti and
Dynes (1974). It is found that the measured speed of propagation as a function of temperature
−2
= A+BT n where A, B,
is well fitted by an empirical law of the form (Coleman et al 1982) νeq
and n are constants. For NaF, some values of A, B, and n giving a good fit are A = 9.09×10−12 ,
B = 2.22 × 10−15 , n = 3.1, when the velocities are measured in cm s−1 and the temperature
in K; for Bi, the following values have been obtained: A = 9.07 × 10−11 , B = 7.58 × 10−13 ,
n = 3.75. The temperature ranges in which the heat pulse propagations have been observed
are 10 < T < 18.5 K (for NaF) and 1.4 < T < 4 K (for Bi). Another relevant quantity is the
heat capacity ceq which varies with T according to ceq = εT 3 . The constant ε depends on the
nature of the crystals: for NaF, ε = 23 erg cm−3 K−4 and for Bi, ε = 550 erg cm−3 K−4 .
Our objective is to determine the values of the parameters a(T ) and γ for NaF and Bi
from the experimental values of A, B, n and ε. After the above experimental expression for
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D Jou et al
Figure 4. Parameter a(T ) expressing the dependence of internal energy with respect to the heat
flux (see equation (10.30)) for NaF and Bi. The intersection of the curves with the T axis indicates
the value of the maximum temperature T ∗ beyond which second sound will not be observed in the
respective materials.
−2
νeq
is used in (10.29), one obtains
A + BT n
τ1
=
+ γ (A + BT n ).
(10.32)
2
λT
εT 5
From the definition (10.21) of a, it is easily checked that
5−n
5A −4 1
a=
(10.33)
BT n−4 +
T − nγ BT n+1 .
2ε
2ε
2
By setting γ = 0, one recovers the result of Coleman and Newman (1988). A further
differentiation of (10.33) with respect to temperature yields
da
1
20A + (n − 5)(n − 4)BT n
− n(n + 1)γ BT n .
(10.34)
=−
5
dT
2εT
2
Since A, B and γ are positive, and since the data for NaF and Bi yield n < 4, it follows
from (10.34) that da/dT < 0. But since—as was argued in the previous section—a must be
non-negative, there exists a maximum temperature T∗ (corresponding to the situation in which
a vanishes) beyond which second sound will not be observed (see figure 4).
This property, which is a prediction of the present model, has indeed received experimental
confirmation: for NaF, T∗ = 18.5 K, and for Bi, T∗ = 4 K. Substituting these maximum values
for T in (10.33) leads to the following expression for γ :
(5 − n)BT∗n + 5A
(10.35)
γ =
nεBT∗n+5
and it is found that γ = 2.7928 × 10−8 (NaF), γ = 2.615 67 × 10−6 (Bi).
The result da/dT < 0 is also of interest because it allows one to determine the critical
value qcrit above which the model is not applicable. Indeed from (10.28) and (10.34), one has
√
2εT 4
qcrit = p
.
(10.36)
20A + (n − 5)(n − 4)BT n + n(n + 1)εγ BT n+5
Extended irreversible thermodynamics revisited (1988–98)
1101
One observes that the presence of non-local effects (γ 6= 0) reduces the value of the critical
bound with respect to that for local theory (γ = 0).
10.5. A generalized minimum entropy production principle
Variational principles have always played a privileged role in physics as mentioned in section 5.
Here, we show a variational formulation of the Guyer–Krumhansl equation in the framework
of the well known Prigogine’s minimum entropy production principle, expressing that purely
dissipative processes (without convection) evolve in such a way that, in the steady state, the
total entropy production is stationary, truly a minimum. Mathematically, this is expressed by
Z
(10.37)
δ σPs dV = 0
wherein δ is the usual variational symbol, and σPs the entropy production derived from classical
irreversible thermodynamics; for heat conduction, one has σPs = qi T,i−1 . It should, however, be
noticed that strictly Prigogine’s principle is only applicable to processes governed by Fourier’s
law with a heat conductivity λ varying as T −2 . In the case of λ proportional to T −1 , Prigogine’s
principle must be changed into δφP (T ) = 0 where φP stands for the total dissipated energy
Z
φP (T ) = T σPs dV .
It is a simple exercise to check that the corresponding Euler–Lagrange equation is qi,i = 0. It
is now shown that the minimum dissipated energy principle still holds when heat transport is
described by the Guyer–Krumhansl steady equations, provided the expression of the dissipated
energy being used is not that derived from classical irreversible thermodynamics, but rather
expression (10.9) obtained in the context of EIT. In the Guyer–Krumhansl approximation,
which corresponds to τ2 = τ3 = 0, expression (10.9) takes the form
s
T σEI
T =
3
3 τN
qi qi +
(qi,j qj,i + 2qi,i qj,j )
τR cν cs2
5 cν
(10.38)
after that identification (10.6) has been used. Moreover, a look at the Guyer–Krumhansl steady
equations
qi,i = 0
1
1
1
qi + cν cs2 θ,i = τN cs2 (qi,jj + 2qj,j i )
τR
3
5
(10.39)
(10.40)
indicates that the energy balance equation (10.39) may be considered as a constraint to
be satisfied by the heat flux vector whose behaviour is governed by equation (10.40):
equation (10.39) plays a role similar to the incompressibility condition in fluid mechanics.
It may be shown that the heat flux that satisfies the Guyer–Krumhansl equation is the one
corresponding to the minimum of dissipated energy:
Z
s
(10.41)
δ T σEI
T dV = 0
s
wherein σEI
T is given by (10.38).
To prove this result, we shall introduce a Lagrange multiplier γL to take into account the
constraint imposed by (10.39). Consider the integral
Z
Z 3 τN
3
q
q
−
q
q
+
q
q
(10.42)
dV
−
γL qi,i dV
φ(q) =
i i
i,j j,i
i,i j,j
2τR cν cs2
10 cν
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D Jou et al
and determine the necessary condition for φ to be stationary under arbitrary variations of qi ,
when qi takes prescribed values (δqi = 0) at the boundaries. Calculation of the first variation
of (10.42) and use of the divergence theorem leads to
Z 3 τN
3
q
−
(q
+
2q
)
+
γ
−
q
δγ
δφ =
δq
(10.43)
i
i,jj
j,j i
L,i
i
i,i
L dV .
τR cν cs2
5 cν
Clearly, δφ = 0 under the conditions that qi,i = 0, and
3
3 τN
qi + γL,i −
(qi,jj + 2qj,j i ) = 0
τR cν cs2
5 cν
(10.44)
which is nothing but the Guyer–Krumhansl equation after the Lagrange multiplier has
been identified with the absolute temperature T . The minimum property of the variational
principle (10.43) is evident, as the dissipated energy is a quadratic form. The physical meaning
of the functional φ is thus clear as it represents the dissipated energy calculated in EIT, and it
thus reinforces the physical role of the generalized entropy.
10.6. Other topics
We end this section by mentioning situations where the Cattaneo equation and some of its
generalizations have found successful practical applications.
(1) High local heating rate. An example of this is the short-pulse laser heating of
metals, which is used in the fabrication of microstructures, synthesis of advanced materials,
measurement of thin-film properties and the analysis of structure transformations. The duration
of these short laser pulses ranges from nanoseconds to femtoseconds and makes EIT a natural
candidate for investigating these short-time processes. These heating rates are comparable
to the thermalization time required by electrons to exchange energy with the lattice and also
with the relaxation time needed by electrons to change their state. This problem has been
treated by using Cattaneo’s equation by Qiu and Tien (1992, 1993), Marciak-Kozlowska and
co-workers (Marciak-Kozlowska and Kozlowski 1996a, b, Marciak-Kozlowska et al 1995)
and Tan and Yang (1997). Other examples of high heating rates are high-speed grinding,
high-speed friction, and explosions.
(2) Fast motion of the heating source. The rate of heating increases with the speed of
the moving heat source. The various effects arising from fast motion have been examined
by Tzou in a series of papers summarized in his monograph (Tzou 1997). The swinging
phenomenon of temperature during the transition of the thermal Mach number (speed of the
source divided by the speed of thermal waves) from the subsonic to the supersonic range, the
physical mechanisms of thermal shock formation, and the local heating induced by dynamic
crack propagation in the transonic regime are three examples which cannot be depicted by the
Fourier law but which have been satisfactorily accounted for by generalized heat equations.
(3) Fast-moving interfaces. Fast motions of interfaces, at a speed higher than the diffusion
speed, are found in rapid solidification of undercooled alloys, or ultrafast melting of pulsedlaser irradiated materials; such topics have a wide spectrum of applications in the science
of new materials. These motions produce significant deviations from local equilibrium at
the solid–liquid interfaces, leading to solute trapping and interfacial undercooling below the
equlibrium temperature. Rapid solidification has been investigated by Sobolev (1991, 1995a, b,
1996a, b), and Galenko and Sobolev (1997) by using Maxwell–Cattaneo equations for the heat
flux and mass flux. They found that, if a flat solidification front moves at a speed higher than
the diffusive speed, the form of the front is stable and the alloy solidifies homogeneously; on
the other hand, when the front speed is slightly lower than the diffusion speed, the flat form
becomes unstable and the solidification front adopts a dendritic profile.
Extended irreversible thermodynamics revisited (1988–98)
1103
(4) Two-temperature models. Up to now, we have only considered relaxational effects
described by one single temperature. However, many practical applications involve systems
consisting of several subsystems, each of which being assigned its own temperature. These
subsystems are, for instance, the electrons and the lattice in a metal submitted to a shortpulse laser heating, where the electron temperature is much higher than the lattice temperature
during a short time. This situation may be described by the following evolution equations for
the electron and lattice temperatures Te and Tl , respectively (Tzou 1997):
∂Te
(10.45)
= ∇ · (λ∇Te ) − C(Te − Tl )
ce
∂t
∂Tl
(10.46)
= C(Te − Tl )
cl
∂t
with C the constant electron–phonon coupling term which accounts for the energy transfer
from the electrons to the lattice, and ce and cl the specific heats of electrons and lattice per
unit volume, respectively. Equation (10.46) yields Te = Tl + (cl /C)∂Tl /∂t. Introducing this
expression into (10.45) leads to
cl ∂∇ 2 Tl
cl + ce ∂Tl ce cl ∂ 2 Tl
.
(10.47)
=
+
∇ 2 Tl +
C ∂t
λ ∂t
λC ∂t 2
This equation may also be obtained by eliminating qi between the energy balance equation
and Guyer–Krumhansl equation under the following identifications: 9τN /5 = cl /C, 3/τR c2 =
(cl + ce )/λ, 3/c2 = ce cl /λC.
It is interesting to note that an equation of the form (10.47) may also be directly derived
from the following constitutive equation, also known as the dual-phase-lag equation (Tzou
1997):
0 ∂T,i
(10.48)
τ1 q̇i + qi = −λ T,i + τ
∂t
which takes into account relaxation effects in both the heat flux and the temperature gradient
(see also equation (4.4)). However, such a procedure yields a parabolic equation, while by
starting with the heat flux and the flux of the heat flux as independent variables as in section 10.1,
one is led to a hyperbolic equation which contains the parabolic equation (10.47) in the limit
of a vanishing relaxation time of the flux of the heat flux.
A two-temperature description is convenient in other systems such as heterogeneous
systems where the liquid and solid phases are at different temperatures; in polyatomic gases,
one can ascribe different temperatures for translational and internal degrees of freedom. These
situations, and the derivation of the suitable generalized equations for heat transfer, have been
studied by Qiu and Tien (1992, 1993), Sobolev (1991, 1993, 1995a, b) and Tzou (1997).
Many other subjects have been explored, both from a theoretical and a practical point
of view. For instance, observations of thermal waves were reported by Mitra et al (1995) in
processed meat, while numerical simulations were performed by Volz et al (1996); applications
of the telegrapher equation to IC chips were investigated by Xu and Guo (1995) and to thermal
laser stereolithography by Luzzi et al (1997b), and general mathematical properties of the
telegrapher equations have been studied by Nagy et al (1994, 1997), Tang and Araki (1996),
Wall and Olson (1997), Barletta (1996) and Barletta and Zanchini (1996). Nonlinear heat waves
have been considered by Orlov (1991), Vázquez et al (1995), Nettleton (1996c) and Méndez
(1997); microscopic analyses of the second sound in photoinjected plasma in semiconductors
were performed by Vasconcellos et al (1995d); analyses of shock waves and second sound in
rigid heat conductors are found in Ruggeri et al (1990, 1996).
A quantized version of the hyperbolic heat transport equation has been proposed by
Marciak-Kozlowska and Kozlowski (1996b, c): the heat wave is quantized in analogy with the
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D Jou et al
quantization of the elastic waves which yields phonons in solid state physics. These authors
have formulated the corresponding equation for a Planck gas, which may be of interest in the
analysis of thermal fluctuations in the primordial universe.
For a wide bibliography see Jou et al (1992, 1996b, 1998) and the website
http://circe.uab.es/eit.
11. Polymer solutions
Polymeric solutions and suspensions are among the applications wherein EIT has found
a significative impetus in recent years, both from the macroscopic and the microscopic
perspective. This is not surprising, because in these systems the viscous pressure tensor
has a relaxation time of the order of several seconds or even of several minutes and therefore
memory effects are easily observable and have important consequences; furthermore, polymers
and suspensions have many practical applications in industry and in biology.
It should also be added that other theories, like those based on internal variables, have
a long tradition in the modelling of rheological materials (Fabrizio et al 1989, Verhás 1997,
Maugin and Muschik 1994). As recalled in section 4.1, EIT departs from these approaches in
several respects.
The contributions of EIT in rheology follow three main lines:
(a) macroscopic analyses based on the evolution equation for P ν , with application to classical
rheological problems;
(b) microscopic approaches in view of a better interpretation of the non-equilibrium entropy
in dilute polymer solutions;
(c) analyses of equations of state under shear flow, with special attention focused on their
effects on the phase diagram of polymer solutions under shear.
11.1. Macroscopic analysis
In this section, a linear viscoelastic model is proposed; it is inspired by the well known Rouse–
Zimm model, which regards the polymer macromolecules as being formed by a chain of N
point-mass beads connected by N − 1 Hookean springs. The forces acting on each bead are of
different kinds: the hydrodynamic drag force due to the motion through the polymeric solution,
the Brownian force caused by thermal fluctuations, the intramolecular force exerted on one
bead through the spring, and external forces. The hydrodynamic force is proportional to the
difference between the bead velocity and the centre-of-mass velocity of the solution. In the
Rouse model, it is admitted that the motions of the N beads are uncorrelated, i.e. one neglects
the hydrodynamic interactions amongst them, whereas this interaction is accounted for in the
Zimm model. The Rouse and Zimm models are appropriate for describing dilute polymer
solutions and were primarily used in linear viscoelasticity whereas, at higher densities, more
sophisticated models, such as the reptation model, are more appropriate.
It is supposed that the roles of the solvent and the polymer can be separated. Accordingly,
the viscous pressure tensor is decomposed into two parts, P ν = P0 + Pp , where subscripts
0 and p denote the contribution of the solvent and the polymer, respectively; next, Pp is
decomposed into N normal modes,
Pp =
N
X
α=1
Pα
(11.1)
Extended irreversible thermodynamics revisited (1988–98)
1105
and each individual mode α is assumed to satisfy, in the linear regime, a relaxational dynamics
of the form
∂ Pα
(11.2)
+ Pα = −2ηα V
τα
∂t
wherein τα is the relaxation time and ηα the viscosity corresponding to the αth normal mode
and V is the symmetric part of the velocity gradient tensor.
In the Rouse model, one can identify the coefficients τα and ηα given by τα = ξ/2H aα
and ηα = nkB T τα , with kB the Boltzmann constant, T the absolute temperature, n the number
of molecules per unit volume, ξ the friction coefficient related to the drag force experienced
by the bead, H the Hookean spring constant, and aα the eigenvalues of the Rouse or modified
Rouse matrices (Bird et al 1987, Doi and Edwards 1986).
Instead of considering the viscous pressure tensor as a single variable, the set of
independent variables must include the contributions arising from the solvent and the N normal
modes of the polymer. In this way, the entropy will depend on pα , the bulk part of the pressure
0
tensor, and P α (α = 0, 1, . . . , N, where α = 0 indicates the solvent), its deviator, besides
the usual variables u (internal energy) and v (specific volume). The polymer concentration is
supposed to remain constant and therefore has not been introduced into the set of variables.
The generalized Gibbs equation then takes the form (Pérez-Garcı́a et al 1989, Lebon et al
1988, 1990, 1991)
ds =
N
N
0
p
1 X
1 X
τα0
τα 0
1
du + dv −
pα dpα −
P α : dP α
T
T
ρT α=0 ζα
ρT α=0 2ηα
(11.3)
provided we neglect the coupling between the various modes.
Up to second-order terms in the velocity gradients and pressure tensors, the entropy
production can be straightforwardly calculated:
X
N
N 0
X
0
τ0 ∂
τα ∂ 0
T σs = −
pα ∇ · ν + α pα −
Pα : V +
P α . (11.4)
ζα ∂t
2ηα ∂t
α=0
α=0
By assuming a linear relation between fluxes and forces, one obtains the following evolution
equations:
∂pα
(α = 0, 1, . . . , N)
(11.5a)
+ pα = −ζα ∇ · ν ,
τα0
∂t
0
0
∂P α 0
τα
+ P α = −2ηα V ,
(α = 0, 1, . . . , N).
(11.5b)
∂t
In most problems the solvent is Newtonian, which means that τ00 τα0 and τ0 τα (α =
1, 2, . . . , N ), so that α runs from 1 to N . Relations (11.5b) are identical to the Rouse–Zimm
equations under the condition that the identifications τα = ξ/2H aα and ηα = nkB T τα are valid.
It is thus seen that EIT provides a very suitable macroscopic framework for the description
of dilute polymer solutions in the domain of validity of the Rouse–Zimm approach. Other
equations more general than (11.2), where the partial time derivative is substituted by more
general time derivatives (corotational, upper convected, etc) are also possible (Bird et al 1987).
Rodrı́guez et al (1988a, b, 1995) have used these equations, together with the
corresponding relaxational equations for the heat flux, to analyse depolarized light scattering in
polymeric solutions. These authors have also proposed a thermodynamic approach to nonlinear
viscoelasticity including heat relaxation and anisotropy (Rodrı́guez et al 1988a); their model
is equivalent to an Oldroyd 8-constant model and predicts a coupling between the heat flux,
the stress tensor and the velocity gradient, with the occurrence of thermal stresses and an
1106
D Jou et al
anisotropic thermal conductivity. Camacho and Jou (1991) have shown that evolution equations
of the Jeffreys type may be obtained thoroughly by including as independent variable a thirdorder tensor describing the flux of the viscous pressure tensor.
11.2. Microscopic foundations
The microscopic foundations of the non-equilibrium entropy have been analysed from
microscopic models for dilute solutions by Camacho and Jou (1990a), Jou et al (1996a) and
Dauby and Lebon (1990, 1991). We briefly summarize the main steps of the approach.
To calculate the polymer contribution to the entropy, we start from the Boltzmann
expression
Z
(11.6)
ρs = −nkB f ln f d0
where f is the polymer distribution function and d0 the volume in the phase space consisting
of the positions r1 , . . . , rN and the velocities ṙ1 , . . . , ṙN of the N beads forming the chain. It is
common to take as variables the centre-of-mass coordinate vector rc and the relative positions
vectors Q1 , . . . , QN −1 , with Qi starting from particle i and ending at particle i + 1. It is also
usual to split the distribution function f into two factors, one depending only on the velocities
and the other only on the configuration:
f = 4(ṙc , Q̇1 , . . . , Q̇N −1 )9(rc , Q1 , . . . , QN−1 ).
(11.7)
Under the assumptions that the distribution of polymers is uniform, the velocity gradient
homogeneous and external forces are position independent, the configuration distribution
function 9 may be expressed as nψ(Q1 , . . . , QN−1 ), with ψ normalized to unity (Bird et al
1987). Moreover, it is supposed that the velocity-dependent part of the distribution function
is given by the Maxwell–Boltzmann local-equilibrium distribution function, i.e. that all the
non-equilibrium effects are due to changes in the configurational distribution.
In terms of the equilibrium distribution function feq , expression (11.6) may be written as
Z
Z
(11.8)
ρs = −nkB f ln feq d0 − nkB f ln(f/feq ) d0.
Since the non-equilibrium effects arise from changes in the configurational distribution
function, the non-equilibrium contribution to s may be written as
Z
u − ueq
− nkB ψ ln(ψ/ψeq ) dQ1 , . . . dQN−1 (11.9)
ρs − ρseq (ueq ) = ρ
T
R
with ρseq (ueq ) = −nkB feq ln feq d0. It should be noticed that in the kinetic theory of
monatomic ideal gases, one has u = ueq and the first term on the right-hand side of (11.9) is
absent.
Instead of the relative position vectors Q1 , . . . , QN−1 , it is usual to introduce the normal
coordinates Q0 , . . . , Q0N −1 of the chain variables, which allow for a factorization of ψ, in such
a way that it simply reads (Bird et al 1987)
ψ(Q01 , . . . , Q0N −1 , t) =
N−1
Y
ψj (Q0j , t)
(11.10)
j =1
where ψj is the distribution function corresponding to the normal mode j . In equilibrium, ψj
is given by the Boltzmann form of the distribution function
3/2
H
H
0
0
0
exp −
Q · Qj
(11.11)
ψj eq (Qj ) =
2πkB T
2kB T j
Extended irreversible thermodynamics revisited (1988–98)
1107
with H the spring constant. The non-equilibrium correction to ψj may be derived from its
evolution equation which, in the absence of external forces, is (Bird et al 1987)
(
)
kT aj ∂ψj
aj c
∂ψj
∂
0
− F j ψj .
= − 0 · (∇ ν ) · Qj ψj −
(11.12)
∂t
∂ Qj
ξ ∂ Q0j
ξ
Fjc is the Hookean force acting between the beads (in our case Fjc = −H Q0j ), ξ is the friction
coefficient describing the hydrodynamic drag force on a bead, and aj are the eigenvalues of
the Rouse matrices. Under a given velocity gradient, the steady solution of (11.12) is
ξ
V : Q0j Q0j
ψj = ψj eq 1 +
(11.13)
4aj kB T
where V is the symmetric part of the velocity gradient. When this result is introduced into the
expression (11.9) for the entropy, one obtains for a plane Couette flow with shear rate γ̇
ρs(u, γ̇ ) − ρseq (ueq ) = 21 nkB
N−1
X
(τj γ̇ )2
(11.14)
j =1
where τj = (ξ/2H aj ); such an expression exhibits explicitly the dependence of s on γ̇ . Note,
furthermore, that the internal energy at equilibrium and at a given temperature differs from the
internal energy under shear at the same temperature, because under shear the polymer stores
an elastic energy uel as a result of the stretching of the chains, which is easily evaluated to be
ρ(u − ueq ) = nkB T
N
−1
X
(τj γ̇ )2 .
(11.15)
j =1
Combining (11.14) and (11.15) results in
ρs(u, γ̇ ) − ρseq (ueq ) = ρseq (u) − 21 nkB
N−1
X
(τj γ̇ )2 .
(11.16)
j =1
It is interesting to compare this expression with its macroscopic equivalent given by
ρs(u, γ̇ ) = ρseq (u) −
1
4
N
X
τj 0 ν 0 ν
P : Pj.
ρηj T j
j =1
(11.17)
The comparison is rather direct since a microscopic evaluation shows that ηj = nkB T τj .
This may be shown, for instance, by starting from the Kramers expression of the polymeric
contribution to the viscous pressure tensor (Bird et al 1987)
0
P ν = −nH
N
−1
X
hQ0j Q0j i + (N − 1)nkB T U
(11.18)
j =1
with h. . .i standing for an average over the configuration space. Combination of (11.18)
and (11.12) yields, in the linear approximation,
∂ 0ν
P = −2nkB T τj V .
(11.19)
∂t j
Comparison of (11.19) with (11.5b) allows us to identify the viscosity related to each partial
viscous pressure tensor as ηj = nkB T τj . With this identification in mind and since in a plane
0
P νj + τj
0
0
Couette flow Pjν : Pjν = 2ηj2 γ̇ 2 , it is seen that the microscopic (11.16) and macroscopic (11.17)
expressions are identical. A similar analysis may be carried out for other models: for instance,
for rigid, elastic, or FENE dumbbells (Camacho and Jou 1990a, b).
1108
D Jou et al
11.3. Phase diagrams under shear flow
The presence of a shear may drastically modify the phase diagram and the value of the critical
point in polymer solutions; indeed a shift of the order of 10 K in the critical temperature is
observed in some solutions under relatively low values of shear viscous pressure (RangelNafaile et al 1984, Wolf 1984, 1991, Criado-Sancho et al 1994, 1998, Jou et al 1995, Onuki
1997). These effects have practical interest in polymer processing, where the homogeneity of
the material must be carefully assessed, and where one can no longer rely on the equilibrium
phase diagram.
At equilibrium, the free energy of mixing is given by the well known Flory–Huggins
formula,
(11.20)
(RT )−1 (1G)F H = n1 ln(1 − φ) + n2 ln φ + χN φ(1 − φ)
where G is the free energy per unit volume, φ the volume fraction of the polymer, and n1 and
n2 the number of moles per unit volume of the polymer and the solvent, respectively; N is
given by N = n1 + mn2 , with m the ratio of the molar volume of the polymer and the molar
volume of the solvent, so that φ = n2 m[n1 + mn2 ]−1 . The interaction parameter χ in (11.20) is
assumed to depend on the temperature according to χ = 21 + ψ[(2/T ) − 1], with ψ a constant
and 2 the so-called theta temperature of the quiescent solution.
In non-equilibrium, the Gibbs expression (11.3) suggests we add to (11.20) a corrective
term of the form
ν 2
)
(11.21)
Gf = v1 NJ (P12
with v1 the molar volume of the solvent. Note that, in general, J , the steady-state compliance
ν 2
) and must be determined from experiments or
defined as τ/η, may depend on φ and (P12
from a microscopic model.
The chemical potential of the ith component is defined as usual by
∂G
∂G
∂φ
=
(11.22)
µi =
∂ni T ,p,Z
∂φ
∂ni T ,p,Z
with Z a non-equilibrium parameter (either the shear rate or the viscous pressure depending
on the external constraints acting on the system). Different choices of Z lead to different
expressions for the chemical potential.
From (11.20) and (11.21) it is shown that the changes in the chemical potentials of the
solvent and the polymer due to mixing are
1µ0
µ0 f
= ln φ + (1 − m)(1 − φ) + χm(1 − φ)2 +
RT
RT
(11.23)
µ1 f
1
1µ1
2
= ln(1 − φ) + 1 −
φ + χφ +
.
RT
m
RT
In equations (11.23) the subscripts 0 and 1 refer to the solvent and the polymer, respectively;
use was made of ∂φ/∂n0 = −(φ/N) and ∂φ/∂n1 = (m/N )(1 − φ), and µ0 f and µ1 f are
the non-equilibrium contributions to the change of the chemical potential due to mixing, i.e.
µif = (∂1Gf /∂ni )T ,p,Z .
The spinodal line in the T , φ plane is given by the relation
∂1µ1
=0
(11.24)
∂φ T ,p,Z
and the critical point, corresponding to the maximum of the spinodal line, is specified by the
supplementary relation
2
∂ 1µ1
= 0.
(11.25)
∂φ 2 T ,p,Z
Extended irreversible thermodynamics revisited (1988–98)
1109
From (11.23) one obtains for the spinodal line and the critical point
∂µ1f
1
∂1µ1
1
1
=−
+ (1 − m−1 ) + 2χφ +
=0
RT ∂φ 1−φ
RT ∂φ
(11.26)
∂ 2 µ1f
∂ 2 1µ1
1
1
1
+
2χ
+
=
−
=
0.
RT
∂φ 2
(1 − φ)2
RT
∂φ 2
The coordinates of the critical point of the quiescent solution are derived from (11.26) by
setting µ1f = 0; this leads to φc = (1 + m1/2 )−1 ; χc = 21 (1 + m−1/2 )2 . When m is very high
(high molecular mass), φc tends to zero and χc to the value 21 , i.e. the critical temperature Tc
tends to the theta temperature.
The shear contribution µ1f modifies the positions of the critical point, the spinodal line,
and the coexistence line. To obtain them, explicit expressions for J as a function of φ and
ν
are needed; these have been established by Rangel-Nafaile et al (1984) for polystyrene
P12
in dioctylphtalate (PS/DP) and by Wolf (1984) for polystyrene in transdecalin (PS/TD) on
experimental bases. For polystyrene in dioctylphtalate, the experimental shifts of the critical
ν
= 100 N m−2 , 14 K at
temperature with respect to the quiescent solution are 4 K at P12
ν
ν
= 200 N m−2 , and 24 K at P12
= 400 N m−2 . Since the temperature shifts are important,
P12
it is clear that equilibrium thermodynamics is inadequate for describing polymer solutions in
the presence of a shear. In the above example, the critical temperature is enhanced by the
shear flow, but this is not a completely general rule. Wolf (1984) has observed that for lowmolecular-weight polymers there is a decrease in the critical temperature, whereas for high
molecular weights there is an increase.
In principle, the equations of state for real polymer solutions must be found experimentally:
however, we are faced with an important lack of information and, on the other hand, the
analytical fit of the experimental data is not an easy task. For these reasons, we shall illustrate
the previous considerations by using the well known Rouse–Zimm model (Bird et al 1987),
which provides a satisfactory description at low values of the shear rate, and we refer to more
specialized papers (Criado-Sancho et al 1991, 1992) for more realistic expressions.
In the Rouse–Zimm model, the steady-state compliance J is given by (Bird et al 1987,
Criado-Sancho et al 1995)
ηs 2
CM2
(11.27)
1−
J =
cRT
η
where c is the polymer concentration expressed in terms of mass per unit volume, η the viscosity
of the solution depending generally on the concentration c and C a parameter whose value will
be discussed below.
The dependence of η with respect to c is crucial in the analysis of the shift of the critical
point: predictions based on the hypothesis that J ∼ c−1 yield unsatisfactory results, because
they lead to a decrease rather than to an increase of the critical temperature (Rangel-Nafaile
et al 1984). It is usually assumed that η(c) takes the form (Wolf 1984, Criado-Sancho et al
1995)
η
= 1 + [η]c + k[η]2 c2
(11.28)
ηs
where k is the so-called Huggins constant and [η] the intrinsic viscosity of the solution.
For the system PS/TD with a polymer molecular mass of 520 kg mol−1 , k = 1.40 (Wolf
1984), [η] = 0.043 m3 kg−1 and the solvent viscosity ηs = 0.0023 Pa s. With these values,
equation (11.28) describes fairly well the viscosity in a concentration range for which the
product c[η] is less than one. Since the critical density of the system is found in this range,
this limitation is not too restrictive.
1110
D Jou et al
Although the expression (11.27) of J is the same in both the Rouse and the Zimm models,
the value of the parameter C is different: in the Rouse model, C = 0.4; in the Zimm model,
C = 0.206. A comparison with experimental data shows that there is a weak shift from the
Zimm to the Rouse model with increasing polymer concentration, which may be described by
assuming that C is a function of the composition, via a reduced concentration c̃ defined by
c̃ = [η]c. Since the functional dependence C = C(c̃) is unknown, it has been proposed to
describe the gradual transition from the Zimm to the Rouse behaviour by means of (CriadoSancho et al 1995)
0.194α c̃
(11.29)
C(c̃) = 0.206 +
1 + α c̃
where α is a parameter which measures how steep the transition is. The corresponding
equilibrium and non-equilibrium spinodal lines are shown in figure 5 for a solution of
polystyrene of molecular mass 520 kg mol−1 in transdecalin. Although at constant γ̇ the
presence of a shear lowers the critical temperature, when the same system is submitted to a
ν
, the critical temperature is increased (see figure 5). Another interesting result
constant P12
derived from figure 5 is that when hydrodynamic interactions are neglected, as in the Rouse
model, the shift of the critical point is maximum; it is minimum in the Zimm model. A
detailed analysis of the theoretical predictions of the temperature shifts for polystyrene in
ν
ν
= 100 N m−2 , 3.5 K at P12
= 200 N m−2 , and 9.2 K
dioctylphatalate gives 1.6 K at P12
ν
−2
at P12 = 400 N m , which are two to three times lower than the actual values mentioned
above (Criado-Sancho et al 1991, Jou et al 1995). This discrepancy may be reduced (CriadoSancho et al 1997) when one takes into account that the dynamical effects corresponding to
the enhancement of fluctuations under the action of the shear flow could contribute to this shift
by an amount comparable to that of the purely thermodynamic effects (see figure 6 and the
discussion below), thus leading to a better agreement with experimental results.
Indeed, the thermodynamic stability relations cannot be taken for granted in nonequilibrium steady states, but they must be justified from a dynamical point of view. For
this reason, several authors (Helfand and Fredrickson 1989, Onuki 1997) have proposed a
dynamical analysis which takes into account the effects of the shear gradient on the density
fluctuations. According to these authors, the increase of turbidity observed in the experiments
would not be due to fluctuations of thermodynamic origin, but to an enhancement of the
fluctuations of purely dynamical origin due to the flow. It should be mentioned that the
range of situations where the thermodynamic criterion of stability is seen to coincide with
the hydrodynamic criterion is rather wide (Casas-Vázquez et al 1993). Moreover, an analysis
of the structure factor, i.e. the Fourier transform of the correlation function of the density
fluctuations, shows that the hydrodynamic effects of the shear rate to the dynamical equations
and the thermodynamic effects arising from the non-equilibrium equations of state are not
contradictory, but that they result in additive contributions to the shift of the spinodal line
(Criado-Sancho et al 1997). In figure 6, the structure factor at wavevector k = 1 and at
angle of 42◦ with the direction of the flow is represented as a function of temperature; the
different curves have the following meanings: curve (1) gives the structure factor at equilibrium
(i.e. at vanishing shear rate); curve (2) includes the dynamical contributions due to the shear
flow; curve (3) shows the purely thermodynamic effects due to the P ν : P ν contribution
to the entropy studied in this section; curve (4) includes both dynamic and thermodynamic
effects; curve (5) includes the additional thermodynamic effects of a coupling term of the
form J · P ν · J to the entropy. At k = 1, dynamical and thermodynamical effects have the
same order of magnitude, but for lower values of k thermodynamic effects are dominant. Of
course, a hydrodynamic description is helpful, as it contains information on the dynamics
of the phase separation and the form of the dispersed phase; nevertheless, a non-equilibrium
Extended irreversible thermodynamics revisited (1988–98)
1111
Figure 5. Spinodal curve of the binary solution PS/TD at equilibrium (Flory-Huggins) and in a nonν
= 150 Nm−2 . The figure collects
equilibrium steady state at constant shear viscous pressure P12
the results corresponding to the Rouse (no hydrodynamic interactions), the Zimm (hydrodynamic
interactions) and two intermediate values (α = 0.5 and α = 1.5) of the parameter α defined
in (11.29). Since the critical temperature corresponds to that of the maximum of the spinodal
lines, it is seen that the larger the hydrodynamic interactions, the smaller the shift of the critical
temperature (Criado-Sancho et al 1995).
thermodynamic description may be interesting because it enlarges the domain of validity
outside equilibrium.
12. Non-Fickian diffusion
Inertial effects and coupling with the viscous pressure tensor seen in (2.23) and examined in
detail in several sections of this review are not only present in heat transport, but also in mass
diffusion, leading to relevant differences with respect to the classical Fick law. Departures
from this law are important in several ways: by the appearance of inertial effects, by couplings
of diffusion to viscosity, or by longitudinal diffusion as in Taylor’s dispersion. Here, we will
show that EIT offers a suitable framework for treating these effects, which are of primary
interest in practice. Indeed, the coupling of diffusion to viscous pressure is important in the
process of diffusion of small molecules into a polymer matrix, and in shear-induced diffusion
in homogeneous polymer solutions. Inertial effects are relevant in some transient regimes,
because of the long relaxation times of polymers and glasses. These couplings cannot be
ignored, because diffusion plays a rate-controlling role in many industrial and biological
processes, as for instance in the dry-spinning of fibres, coating of substrates via deposition and
subsequent drying of polymer solutions, and in the design of drug-delivery systems.
The evolution equations for the diffusion flux, the bulk viscous pressure and the pressure
tensor are analogous to (2.37)–(2.39) with the condition of replacing the heat flux by the mass
1112
D Jou et al
Figure 6. Structure factor S̃ for a transdecalin-polystyrene solution at k = 1 and at an angle
of 42◦ with the direction of the flow velocity. The vertical asymptote to each curve gives the
apparent critical temperature. See the text for a detailed explanation of the meaning of each curve
(Criado-Sancho et al 1997).
flux J . They take the form (Jou et al 1991, Pérez-Guerrero and Garcı́a-Colı́n 1991).
0
τ1 J˙ = −(J + D∇c) + β 00 D̃T ∇ · P ν + β 0 D̃T ∇pν
τ0 ṗ ν = −(pν + ζ ∇ · ν ) + β 0 T ζ ∇ · J
0
0
(12.1)
(12.2)
0
τ2 (P ν ). = −(P ν + 2ηV ) + 2β 00 T η(∇ 0 J )s
(12.3)
where β 0 and β 00 describe the coupling of diffusion and viscous effects and also appear in the
expression of the entropy flux, which, in the absence of heat effects, is given by
0
J s = −µT −1 J + β 00 P ν · J + β 0 p ν J .
(12.4)
The last two terms on the right-hand side of (12.4) are non-classical and similar to those
introduced in (2.15); the quantity D̃ in (12.1) stands for D̃ = D(∂µ/∂c)−1 , with D the
usual diffusion coefficient and c the concentration. Equations (12.1)–(12.3) clearly exhibit
the couplings between diffusion and viscous pressure. For instance, during diffusion of small
molecules in a polymer matrix, these couplings, resulting from the swelling due to the solvent,
produce a relative motion between neighbouring polymer chains, whose mutual friction may
emerge into a viscous stress. Equations (12.1)–(12.3) have been used for the analysis of
diffusion in polymer solutions (Goldstein and Garcı́a-Colı́n 1993, 1994), in dense fluids of
small molecules (Nettleton 1988a, 1993), and in dynamical effects of shear flows on the critical
point in polymer solutions (Onuki 1997, Helfand and Fredrickson 1989). Expressions (12.1)–
(12.3) have been written in the context of a Hamiltonian formulation by Beris and Edwards
(1994). We deal here briefly with two applications.
Extended irreversible thermodynamics revisited (1988–98)
1113
12.1. Case-II diffusion
Experiments where non-Fickian effects are perceptible are sorption, desorption and permeation
of a solvent of low molecular weight in a membrane of glassy polymer. The initial concentration
of the permeant is suddenly increased (sorption) or decreased (desorption) around the sample,
or at one side of it (permeation). The relevant quantities are the mass uptake per unit surface
as a function of time m(t), in sorption and desorption experiments, or the volume throughout
Q(t) per unit area of membrane, in permeation.
When small solvent molecules penetrate into a polymer matrix, a sharp advancing
boundary is produced between the inner glassy region and the outer swollen gel (Samus and
Rossi 1996, Billovits and Dunning 1994, Cody and Botto 1994, Cairncross and Dunning
1996). The boundary moves with a constant velocity in case-II diffusion, or it accelerates
in super-case-II diffusion. The short-time expression for the mass uptake may be expressed
as m(t)/m(∞) ∼ t n , with n = 1 for case-II, n > 1 for super-case-II, and n = 21 for the
usual Fickian diffusion. Although a well-defined boundary could also appear in classical Fick
diffusion with a sufficiently steep variation of the diffusion coefficient with the concentration,
such a boundary would move as t 1/2 and not as t. Case-II diffusion can be simply described
by means of a Maxwell–Cattaneo equation as obtained from (12.1) with β 0 = β 00 = 0. Supercase-II diffusion cannot be interpreted from the simple Maxwell–Cattaneo model, but requires
the coupling between stress and diffusion which is described by the last term in the right-hand
side of (12.1).
In accordance with the literature, we restrict ourselves to the one-dimensional problem,
ν
as variables, and ignore the bulk effect (pν = 0); moreover, we
with only c, Jx , and Pxx
0
assume that τ0 = τ2 , β = β 00 = β, and that the velocity gradients vanish. With these
simplifications, (12.1)–(12.3) reduce to
∂P ν
∂c
+ β D̃T xx
(12.5a)
τ1 J˙x + Jx = −D
∂x
∂x
∂Jx
ν
ν
(12.5b)
+ Pxx
= T βηl
τ2 Ṗxx
∂x
with ηl = 43 η + ζ the longitudinal viscosity. In many situations it is seen that τ1 in the
evolution equation (12.5a) of the diffusion flux may be neglected in comparison with τ2 ,
which characterizes the relaxational effects for the viscous pressure. When the term in τ1 is
neglected, (12.5a) simplifies as
∂
ν
).
(12.6)
Jx = −D̃ (µ − T βPxx
∂x
This equation is to be related with the basic idea underlying the Thomas–Windle model
(Thomas and Windle 1982), one of the most successful ones of super-case-II diffusion. These
ν
affects the chemical potential in the same way as an ‘osmotic’
authors assume that the stress Pxx
pressure; in other words, diffusion finds its origin in the gradient of a non-equilibrium chemical
potential given by
Z p+Pxxν
∂µ
ν
ν
µ(T , p + Pxx
, c) = µeq (T , p, c) +
dp 0 0 = µeq (T , p, c) + v Pxx
(12.7)
∂p
p
with v the specific volume of the solvent. Comparison of (12.6) with (12.7) yields the
identification −T β = v.
ν
and Jx is needed. This relation
To close the set of equations, a second relation between Pxx
is provided by (12.5b), written in the form
∂Jx
ν
ν
+ Pxx
= −ηl v
.
(12.8)
τ2 Ṗxx
∂x
1114
D Jou et al
In a first approach, Thomas and Windle took τ2 = 0, but this contrasts with some experimental
observations. Later, Durning and Tabor (1986) returned to (12.8) with τ2 6= 0. The
corresponding diffusion flux resulting from the combination of (12.6) and (12.8) is then given
by
Z t
∂c
∂c
∂
Jx = −D
dt 0 exp[−(t − t 0 )/τ2 ] 0
(12.9)
− D 00
∂x
∂x −∞
∂t
after using the mass balance equation ∂c/∂t = −∂Jx /∂x and identifying the coefficient D 00 as
D 00 = v 2 D̃ρηl /τ2 . Durning and Tabor (1986) analysed the consequences of (12.9) in classical
and oscillatory sorption, by taking suitable relations for D(c) and ηl (c) and obtained good
agreement with experimental results. Considering the full model with a non-vanishing τ1 ,
Beris and Edwards (1994) were able to interpret the sorption overshoot which is observed on
some occasions.
12.2. Taylor dispersion
Taylor dispersion describes the longitudinal dispersion of a solute in a solvent flowing along
a rectilinear duct. Taylor showed that the combined action of a gradient in the velocity field
and the transverse molecular diffusion leads, after a relatively long time, to a longitudinal
diffusion, with an enhanced diffusion coefficient given by Def = Dm + DT , with Dm the
molecular diffusion coefficient in a fluid at rest and DT the Taylor correction, which is of the
form DT = AU 2 d 2 /Dm , with U the mean velocity, d the width of the duct and A a numerical
coefficient which depends on the details of the flow (for a plane Couette flow between parallel
1
and for a cylindrical Poiseuille flow in a duct of radius
plates separated a distance d, A = 210
1
d, A = 48 ).
Taylor’s diffusion is important in many circumstances, such as diffusion in rivers and
estuaries, arteries or veins. Furthermore, DT is much larger than Dm , and is therefore more
accessible to direct measurements. Taylor’s formula is only valid for asymptotic long times,
and one has tried to obtain a description valid at any value of time (see Camacho 1993a, b
for references). However, since the direct study of the intermediate timescales turns out to
be rather complicated, Camacho (1993a, b, 1996) proposed a simplified approach based on a
generalized Maxwell–Cattaneo equation for the Taylor diffusion flux JT , taking the form
∂JT
∂JT
∂c
∂ 2 JT
+ τT βu
= −DT
+ lT2
(12.10)
∂t
∂x
∂x
∂x 2
where u is the mean velocity in the direction of the flow, c the concentration and β a
phenomenological coefficient. This relation contains relaxational and non-local terms and
includes, besides, a new term (the third term on the left-hand side) which describes a transient
anisotropic dispersion due to the fact that the solute disperses differently down-flow and upflow. Note that the non-local term may be obtained directly from (12.5) by setting τ2 = 0
in (12.5b) and introducing it in (12.5a).
Substitution of equation (12.11) into the solute balance equation results in
J T + τT
τT
∂ 2c
∂ 3c
∂ 2 c ∂c
∂ 2c
∂ 3c
∂ 4c
2
2
+
τ
−
(τ
+
βu
D
+
l
)
+
τ
βuD
+
l
=
D
D
.
T
T
m
T
m
m
T
∂t 2 ∂t
∂x∂t
∂x 2 ∂t
∂x 2
∂x 3 T
∂x 4
(12.11)
The results derived from this equation have been compared with a numerical simulation of a
flow between parallel plates (Camacho 1993b) (figure 7). In this experience (transmission
dispersion simulation), one computes the average of the displacements of the particles
with respect to the mean convective motion over the simulation distribution h(1x)2 (t)i at
Extended irreversible thermodynamics revisited (1988–98)
1115
Figure 7. Taylor dispersion: comparison between the theoretical predictions of (12.11) (solid
curves) and numerical simulations. Squares and diamonds correspond to transmission and echo
simulations, respectively (Camacho 1993b).
different points as a function of time, and represents the ratio h(1x)2 i(t)/2t = D ∗ (t) versus
the penetration length of the solute as a whole, given by L = hx(t)i; D ∗ is called the
‘effective dispersion coefficient’, as it has the dimensions of a diffusion coefficient and tends
asymptotically to D in the long-time limit. Other simulations (echo dispersion simulations)
consist of letting the system evolve during a time ti and then suddenly reversing the velocity
field and again representing D ∗ (t) as a function of L. The results of the theoretical predictions
based on (12.11) are compared with those of the simulations in figure 7. The agreement is
very satisfactory for a wide range of velocities (always in the laminar regime) and therefore it
provides a useful generalization of Taylor’s results for all time regimes.
The coefficients τT , lT and DT in (12.11) are not arbitrary. They may be obtained from
νn and
an analysis of the behaviour of the Fourier modes of the density, cn (x, t), the velocity
P
the fluxes Jn = 21 cn (x, t)νn , and by relating them to the Taylor flux as JT = n Jn ; it was
found (Camacho 1993a, b) that
τT =
DT
,
hν 2 i
lT2 = τT Dm ,
DT =
∞
X
1 2
ν τ
2 n n
(12.12)
n=0
where the time spectrum τn (τn is the relaxation time of the nth mode Jn of the diffusion flux)
depends on the kind of hydrodynamic flow. The reader interested in further details is referred
to Camacho (1993a, b), who also determines the non-equilibrium entropy and its relation with
the evolution equation for the diffusion flux, in total agreement with EIT.
Let us finally stress that the interplay between the velocity field and diffusion is also
important in fluid transport in porous media (del Rı́o and López de Haro 1991, del Rı́o et al
1992a, b) or in membrane transport (del Castillo and Rodrı́guez 1989, Baranowski 1991)
opening the way to other applications of practical interest for the models described by (12.5)
or (12.10).
1116
D Jou et al
13. Dielectric relaxation of polar liquids
A possible description of dielectric relaxation in polarizable media is to introduce one or several
polarization vectors as independent variables (see, for instance, Ciancio and Restuccia 1990,
Ciancio et al 1990, Ciancio and Verhás 1991, Conforto and Giambò 1996). Several authors
(e.g., del Castillo and Garcı́a-Colı́n 1988, del Castillo and Dávalos-Orozco 1990, del Castillo
et al 1997) have studied the ultrafast dielectric response of dense polar liquids in the framework
of EIT, incorporating some cross-couplings which were neglected hitherto but which may play
a significant role.
The state variables are the density of internal energy ρu and ρp , the density of polarization
charges (defined as minus the divergence of the polarization vector), which are conserved
variables, and two fast and non-conserved variables, Jp = dP /dt, the polarization current
(with P the polarization) and 3, the flux of Jp. , i.e. d2 P /dt 2 . It should be stressed that the
meaning of the fluxes Jp and 3 is conceptually different from that of the fluxes of heat or
mass diffusion introduced earlier. Indeed, the latter refer to displacements of particles (or
quasiparticles) in space, while the former are taken at a given point of space and describe
the variation of a given quantity, respectively P and its time derivative, in the course of
time. It could be asked why one does not introduce fluxes of higher orders, such as dn P /dt n ,
analogously to section 7.3. The continued-fraction formalism analogous to that of (7.30)
would correspond in this case to the Mori (1965) formalism. However, we have seen in
section 10 that one still obtains very reasonable results by taking only the flux of heat and its
respective flux as variables. Therefore, and by analogy, we find it reasonable here to truncate
the hierarchy of variables at the second order in time derivatives. Alternatively, the asymptotic
methods mentioned in section 7.3 could be used to define effective coefficients appearing in the
formulation with two fluxes. As will be commented on later, comparison with experimental
results is rather satisfactory.
The corresponding generalized Gibbs equation may be written as
(13.1)
ρ ṡ = T −1 ρ u̇ + T −1 ψ ρ̇p + (α1 Jp + α2 3) · J˙p + (α3 Jp + α4 3) · 3̇
where α2 = α3 by the integrability condition, and ψ is the local electric potential, related to
the electrical field E by E = −∇ψ. It may be noted in (13.1) that since both Jp and 3 are
vectors, they are coupled through coefficients α2 and α3 . Classical irreversible thermodynamics
corresponds to the particular situation where αi = 0.
The derivatives of u and ρp are given by the balance laws
ρ u̇ = E · Jp ;
ρ̇p = −∇ · Jp .
(13.2)
s
s
Taking for the entropy flux J the expression J = (ψ/T )Jp , it is easily checked that the
entropy production is given by
d Jp
d Jp
1
d3
d3
s
σ =
· Jp + α 2
· 3 (13.3)
(E − E0 ) + α1
+ α3
+ α4
T
dt
dt
dt
dt
with E0 the value of E at equilibrium. In the linear approximation, one may formulate the
evolution equations for the fast variables as follows:
d Jp
d3
1
(E − E0 ) + α1
+ α3
= µ1 Jp + µ3 3
T
dt
dt
(13.4)
d Jp
d3
α2
+ α4
= µ3 Jp + µ2 3
dt
dt
after we have assumed that the Onsager symmetry relations remain valid. The above equations
may be written in terms of P and 3 in the form
d2 P
d
(13.5)
3 − (P − χ0 E ) = 0
−τ1 2 + τ2 δ 1 + λ2
dt
dt
Extended irreversible thermodynamics revisited (1988–98)
1
d
d dP
3+
1 + λ4
=0
1 + λ3
dt
δ
dt dt
1117
(13.6)
where τ1 = −T χ0 α1 , τ2 = −µ23 T χ0 /4µ2 , δ = 2µ2 /µ3 , λ2 = λ4 = −2α3 /µ3 , λ3 = −α4 /µ2
and χ0 is the electric susceptibility at equilibrium. If the relaxation times λ2 , λ3 and λ4 are
small, (13.6) reduces to 3 = −(µ3 /2µ2 ) dP /dt and (13.5) takes the form
d2 P
dP
+ τ2
(13.7)
+ (P − χ0 E ) = 0.
dt 2
dt
This is a first generalization of the Debye equation for P . The latter equation includes only
the first-order derivative of P (but not the second one) and gives good predictions at low
frequencies but not at higher ones.
The autocorrelation function for the polarization vector P is written as
τ1
hδPa (t)δPa (0)i
(13.8)
hδPa (0)δPa (0)i
where a = x or z (x is the transverse mode and z the longitudinal mode, when E is assumed to
be directed along the z-axis). From this function one may directly obtain relevant experimental
quantities. Indeed, the dielectric susceptibility χ(ω) and the complex dielectric constant ε(ω)
are given by the relations
φa (t) =
ε(ω) − ε∞ ε(0)
χ(ω) − χ∞
= L(−φ̇z )
=
(longitudinal mode)
χ(0) − χ∞
ε(0) − ε∞ ε(ω)
(13.9)
ε(ω) − ε∞
χ(ω) − χ∞
=
= L(−φ̇x )
(transverse mode)
χ(0) − χ∞
ε(0) − ε∞
where L(. . .) denotes the Laplace transform, while index ∞ corresponds to the limit of infinite
frequency. From (13.5) and (13.6) it follows that the Fourier transform of (13.8) has the form
φa (ω) =
1
−iω −
M0
M 0 + γ2
−iω +
γ3
−iω −
M0
(13.10)
with M0 = −γ3 /γ1 , γ1 = (λ1 + 2λ2 τD )(λ1 λ3 + λ22 τD )−1 ; γ2 = (λ3 + τD )(λ1 λ3 + λ22 τD )−1 ; and
γ3 = (λ1 λ3 + λ22 τD )−1 . If λ2 = 0, this expression reduces to the classical model of Kivelson
and Keyes (1972). The new parameter λ2 in (13.5) and (13.6) represents the coupling between
two dynamical effects. Del Castillo et al (1997) have focused on the transverse mode of
chloroform in the Cole–Cole diagram. They show that for chloroform at 295 K (ε(0) = 4.71,
ε∞ = 2.13), the choice τ2 = 6.33 ps, τ1 = 0.38 ps, λ3 = 0.33 ps and λ2 = 0.012 ps
yields better agreement for the high-frequency knob of the Cole–Cole plot than the model
corresponding to λ2 = 0. This exhibits the importance of the cross-coupling effects which are
easily taken into account in a macroscopic formalism but which are more difficult to implement
in a microscopic approach. Note that the coupling provided by λ2 (= λ4 ) is analogous to the
coupling between the flux of the heat flux Q and the heat flux q through the terms involving
gradients in the set of equations (10.2)–(10.4). Thus, the general scheme which successfully
describes many non-classical heat transport effects is also able to describe dielectric relaxation
in great detail.
Relaxational phenomena have been inspiring for EIT, because of the nonlinear character
of the decay, which on some occasions must be modelled by a stretched exponential, known
as the Kohlrausch–Williams–Watts law, rather than by the usual exponential decay (Jou and
Camacho 1990, Garcı́a-Colı́n and Uribe 1991, Ramos-Barrado et al 1996). To deal with
1118
D Jou et al
this phenomenology, the dissipative fluxes can no longer be considered as entities with one
single relaxation time, but rather as a sum of several contributions, each with its characteristic
relaxation time, all of them being related by some scaling law (Jou and Camacho 1990).
Another approach is to use nonlinear fractional derivatives (Nonnenmacher and Nonnenmacher
1989). These results are worth stressing as they indicate that EIT encompasses a much wider
range of dynamical laws than the original version exclusively dedicated to single exponential
dynamics, but they are beyond the scope of this review.
14. Microelectronic devices
One of the processes which finds a natural place in the framework of EIT is charge
transport in submicronic microelectronic devices, a very important field in modern technology,
which strives for miniaturization and optimization of the devices. These phenomena are
generally described either by Monte Carlo simulations or hydrodynamic formulations. Monte
Carlo analyses are useful, because they start from first principles and are applicable far
from equilibrium; however, they are extremely computer-time and memory consuming.
Hydrodynamic methods take as variables a reduced number of lowest-order moments of the
carriers’ velocity distribution function, such as density, charge flux, kinetic energy, energy flux
and so on, which are easily measurable and controllable. Their numerical solution is much
faster than with Monte Carlo methods, but they meet several fundamental challenges.
14.1. Hydrodynamic description of transport
Indeed, the Boltzmann equation leads to a whole hierarchy of evolution equations for the
moments, which for practical reasons must be truncated at a given order. Depending on
the choice of variables and the level at which the hierarchy is cut, one obtains different
hydrodynamical models. One of the simplest models is the drift-diffusion model (Hänsch
1991), which considers the number density of electrons and holes and their respective velocities
as independent variables, but not their energies. A more sophisticated model is that of Baccarani
and Wordeman (1982), which includes the energy of electrons and holes as further independent
variables, but the heat flux is still viewed as a dependent variable given by the Fourier law.
Introduction of more independent variables requires us to generalize the basic concepts
of hydrodynamics and thermodynamics, which in their classical version are restricted to the
five lowest moments. Clearly, EIT turns out to be a natural candidate for this extension. Anile
and co-workers (Anile and Pennisi 1989, 1992a, b, Anile and Muscato 1995, 1996, Anile
et al 1995, Romano 1996) have applied EIT to submicronic devices beyond the classical
diffusion-convection approximation. The viscous pressure tensor and the heat flux were
selected as independent variables, the criterion imposing the positiveness of entropy production
was introduced to close the hierarchy of higher-order moment equations and Monte Carlo
simulations were used as a benchmark to check the results of hydrodynamic predictions.
The main steps of their approach can be summarized as follows. The evolution equations
for the moments are directly derived from the Boltzmann equation. For electrons in the
conduction band of the semiconductor, one has
∂f
(14.1)
+ ν (k) · ∇f − eE · ∇k f = Q
∂t
with e the charge of the electron, f (x, k, t) the distribution function, k the electron momentum,
ν (k) = ∇k ε the electron group velocity, E the electric field and Q the collision term. In the
effective-mass approximation, the energy ε(k) is given by ε(k) = k 2 /2m∗ , m∗ being the
effective electron mass. After multiplication of (14.1) respectively by k, k · k, k . . . k, . . . ,
Extended irreversible thermodynamics revisited (1988–98)
1119
and integration, one obtains a hierarchy of equations for the different moments. Besides the
mass conservation equation for the zeroth-order moment, one has (Anile and Pennisi 1992a, b)
∂θij neEi
∂
(nνi ) +
+
= Qi
∂t
∂xj
m∗
0
(14.2)
0
0
∂ θ ij ∂ θ ij r 2neEhi νj i
+
+
= Qij
∗
∂t
∂xr
m
(14.3)
R
where n is the density
number of electrons, θij and Qij stand for θij = (1/m∗2 ) dkf ki kj ,
R
and Qij = (1/m∗ ) dk Qki kj and so on, and h. . .i denotes the
R symmetric and traceless part
of theRcorresponding tensor. Furthermore, the energy W = dk f ε(k) and the energy flux
Si = dk f ε(k)νi (k) obey the following relations:
∂W ∂Si
+ neEi νi = Qw
(14.4)
+
∂t
∂xi
∂Si ∂Sij
+
+ e[Ej θij + (W/m∗ )Ei ] = Q0i
(14.5)
∂t
∂xj
R
R
R
where Qw = dk Qε(k), Q0i = (1/m∗ ) dk Qε(k)ki , Sij = (1/m∗2 ) dk f ε(k)ki kj .
These equations must be complemented by the Poisson equation which relates the electrical
potential to the charge density. The constitutive equations for the collision terms are drawn
from the Baccarani–Wordeman model (1982)
nνi
,
Qi = −
τp
Q0i
Si
=− ,
τq
W − W0
Qw = −
,
τw
0
θ ij
Qij = −
τs
0
(14.6)
where τp , τq , τw and τs are the corresponding relaxation times. To recover the Onsager relations
at high and low energies, Anile and Pennisi (1992b) have shown that τp and τq must depend
on the magnitude of the energy flux S and the electric current J .
To close the set (14.2)–(14.5), based on the variables n, νi , W, Si and θhij i (both for
electrons and for holes), one needs constitutive equations expressing θij r , Sij , in terms of the
basic variables. The truncation plays a decisive role and therefore it deserves a careful analysis.
EIT provides a way to determine which kind of truncation is compatible with thermodynamics
(Anile and Pennisi 1992a, b, Anile and Muscato 1995).
Let us decompose k according to k = m∗ (u + c), where u is the average velocity and c
the peculiar electron velocity; θ̂ij r denotes the third-order moments of the distribution function
with respect to c. The part of Sij independent of u is θ̂ij rr and the part of the energy flux
Si independent of u is the heat flux qi . By applying the methods of EIT, Anile and Muscato
(1995) find
1
2m∗
(qi δj k + qj δki + qk δij ) +
(qi θ̂j k + qj θ̂ki + qk θ̂ij )
5
15nkB T
∗
2m
ql (θ̂hlii δj k + θ̂hlj i δki + θ̂hlki δij )
−
(14.7)
15nkB T
∗
5n
2
7kB T
m
=
(kB T )2 + σ −
q · q δij + 2σ qi qj +
θ̂hij i +
θ̂hili θ̂hj li
∗
2m
5nkB T
2
n
θ̂hij ki =
θ̂ij rr
where σ is a free parameter determined from the comparison with computer simulations or
experiments. For silicon at room temperature and considering only intravalley scattering with
acoustic phonons and inelastic intervalley scattering with optical phonons, it was found that
the hydrodynamic model was in good agreement with Monte Carlo simulations (Anile and
Muscato 1996). The errors resulting from the closure at the third- and fourth-order moments
1120
D Jou et al
are less than 7% for electric fields up to 105 V cm−1 for σ = −0.1321 in the stationary and
homogeneous situation. Anile and Muscato (1996) attributed the success of the hydrodynamic
approach to the wide range of validity of the isotropy property of the distribution function,
even in strong electric fields, due to the dominant effect of scattering of electrons with nonpolar
optical phonons, which is elastic and isotropic. Nonstationary and strongly inhomogeneous
situations are being studied.
Another interesting subject in microelectronic devices of nanometric size is the
contribution of ballistic electrons to charge transport, because the size of the device becomes
comparable to the mean-free path. Ballistic transport of electrons and holes was observed in
GaAs in 1985, and advantage has been taken of this characteristic of transport to increase the
speed of transport and improve the efficiency of the devices. A description of ballistic transport
in terms of moments raises some difficulties (Nekovee et al 1992), because a ballistic peak
in the velocity distribution function requires a high number of moments for its description,
instead of only a few moments as in situations close to equilibrium. These difficulties are
analogous to those explored in detail in section 7.3 (Dedeurwaerdere et al 1996), where the
role of higher-order fluxes in the determination of the value of the speed of ballistic thermal
pulses was discussed. To obtain the right value of the ballistic pulses, a high number of fluxes
must be considered, or an asymptotic procedure such as the one proposed in (7.34) must
be used to define suitable effective relaxation times, as in (7.38). Transport properties and
thermal waves in photoexcited plasma in semiconductors has also been investigated by using
the non-equilibrium statistical operator method combined with EIT (Vasconcellos et al 1994,
1995d).
14.2. Nonlinear effects. Flux limiters
Besides microelectronic devices, EIT has also been applied to several electrical transport
phenomena in plasmas (Anile and Muscato 1989, Martı́nez-Romero and Salas-Brito 1991,
Llebot 1992). A generalized Einstein relation between diffusion coefficient and mobility has
been derived for high values of the electric field (González and Jou 1992). This is achieved by
starting from the expression
∂µ
qD
(14.8)
=n
ν
∂n T ,V
with q the ionic charge, D the difusion coefficient, ν the ionic mobility, µ the chemical potential
per particle and n the number of ions per unit volume. The generalized non-equilibrium
chemical potential µ is given by
∂(τ V /2σe ) 2
(14.9)
J
µ = µeq +
∂N
with J the electric current density and σe the electrical conductivity. Introducing in (14.9) the
Drude relation σe = nq 2 τ/m, with m the mass of the ions, and substituting the subsequent
expression for the chemical potential into (14.8), results in
τ 2q 2 2
qD
=1+
E .
(14.10)
νkB T
mkB T
In the limit of vanishing electric field E (14.10) reduces to the classical Einstein relation,
but it differs when the values of E are not negligible. This simple result agrees qualitatively
with more sophisticated analyses based on kinetic theory (Garcı́a-Colı́n and Uribe 1991) or on
information-theoretical methods (Vasconcellos et al 1995d).
Another interesting aspect of nonlinear transport is the saturation of the fluxes at high
values of the thermodynamic forces. For example, a phenomenological expression for the
Extended irreversible thermodynamics revisited (1988–98)
1121
relation between the electric flux Je and the electric field E expressing this behaviour is the
so-called Chauchy–Thomas relation (Hänsch 1991)
σe E
Je = q
(14.11)
1 + σνse E 2
with σe the classical electrical conductivity for low values of E and νs the Fermi speed of
electrons. This expression reduces to Ohm’s law for low values of the electric field, and tends
to a saturation value for Je for high values of the electric field. A flux limiter may be obtained by
generalizing the hierarchy (7.30). Indeed, in the presence of a non-vanishing thermodynamic
force X (as, for instance, E , ∇T , etc), equation (7.30) may be generalized as follows (Zakari
1997):
(14.12)
τn J˙(n) + J (n) = λn ∇ J (n−1) + γn ∇ · J (n+1) + λ0n X · J (n−1) + γn0 X · J (n+1) .
In section 7.3 we have shown how this hierarchy leads to a continuous fraction expansion of
thermal conduction. If we are interested in the response of the system to the external force
X in the steady state situation, rather than in the role of frequency and wavelength, we may
neglect in (14.12) the terms in gradients and time derivative and we obtain
σe E
(14.13)
J=
λ00 γ10 E 2
1−
λ01 γ20 E 2
1−
λ0 γ 0 E 2
1− 2 3
1 − ···
where we have taken X = E , J (0) ≡ 1 and λ00 = σe . In the case when λ00 = λ01 = · · · = λ0n =
σe and γ10 = γ20 = · · · = γn0 = −α > 0, expansion (14.13) simplifies as
σe E
q
J=
.
(14.14)
1
+ 41 + ασe E 2
2
This law yields a saturation value Jsat = (σe α)−1/2 , which coincides with the saturation value
of (14.11) if α is identified as α ≡ νs−1 .
There are other phenomena where flux limiters play a crucial role. Typical situations
arise, for instance, in radiative heat transfer, where the speed of photons is the speed of light
c, and where the maximum heat flux is aT 4 c (Mihalas and Mihalas 1984, Levermore 1984,
Levermore and Pomraning 1981), or in plasma physics, where the speed of electrons is of
the order of (kB T /m)1/2 and the energy density is proportional to kB T , so that the maximum
heat flux will be of the order of kB T (kB T /m)1/2 (Sharts et al 1981). Such bounds play an
important role in laser–plasma interaction in laser-induced nuclear fusion, or in the collapse
of stars. Although this saturation effect cannot be described by the classical Fourier law, it
may be introduced by allowing the thermal conductivity λ(T , ∇T ) to depend not only on the
temperature but also on the temperature gradient. Thus, we have in a steady state
q = −λ(T , ∇T )∇T .
(14.15)
Several different expressions for the form of λ(T , ∇T ) have been proposed. One of the
best known is that derived by Levermore (1984) from a modified diffusion model for photons,
and given by
λ(T , x) = 3λ0 (T )x −1 (coth x − x −1 )
0
0
(14.16)
where x = l ∇T /T , l being a length of the order of the mean free path and λ0 (T ) the thermal
conductivity near equilibrium. For small values of the temperature gradient (x → 0), one
1122
D Jou et al
has λ → λ0 , whereas for high values of ∇T , λ → 3λ0 T (l 0 ∇T )−1 , and the corresponding
saturation value of the heat flux is qmax = 3λ0 T / l 0 . Since l 0 = cτ and λ0 = 13 aT 4 c2 τ , with a
the constant defined by U = aV T 4 , one finds, finally, that qmax = aT 4 c. It is important to note
that if the mean free path l 0 in x tends to zero, the nonlinear law (14.16) reduces to the linear
Fourier’s law. Thus, the introduction of a non-vanishing relaxation time or a non-vanishing
relaxation length in EIT, whose linear consequences we have analysed in much detail in this
review, may also have consequences for the nonlinear transport equations.
EIT has been applied to the analysis of flux limiters in two different circumstances. On the
one hand, Anile et al (1991) have shown that a flux limiter may be obtained by assuming that
there exists a reference frame where the radiation is in equilibrium; in this case, the entropy
of the gas in motion coincides with that derived in EIT. Another analysis by Jou and Zakari
(1995b) is based on the fluctuation-dissipation expression for the thermal conductivity in nonequilibrium steady states. Information theory has been used by Jou and Zakari (1995a), Zakari
and Jou (1995) and Mascali and Romano (1997) to obtain an explicit form for flux limiters
in relativistic gases, whose cosmological consequences have been examined by Bonnano and
Romano (1994).
15. Superfluids
Second sound (undamped thermal waves) was discovered for the first time in liquid helium II the
superfluid phase of liquid helium which is found below 2.17 K. It is well known that helium II
shows very odd effects: a very small viscosity, a high thermal conductivity, a strong coupling
between momentum and heat flow, and the fountain effect whereby a temperature difference
generates a pressure difference. The high thermal conductivity and the long relaxation time
of the heat flux q suggest that this flux should be taken as an independent variable. Analyses
of superfluid helium II by means of EIT may be found in Greco and Müller (1983), Mongiovı̀
(1991, 1992, 1993a, b) Mongiovı̀ and Romeo (1995, 1996), Lindblom and Hiscock (1988) and
Giambó and Lebon (1991). We briefly summarize below the main features.
The classical theory of superfluid helium II is the two-fluid model proposed by Landau,
which regards the system as composed of a normal component with normal viscosity and
nonzero entropy and a superfluid component with zero entropy and zero viscosity. The
latter component, which is absent above the lambda transition temperature, is supposed to
be composed of particles in the condensed Bose–Einstein state, while the normal fluid is
formed by particles in excited states. The expression for the dominant contribution to the heat
flux has the form
q = T sρs (ν n − ν s )
(15.1)
with ν n and ν s the barycentric velocities of the normal and the superfluid components,
respectively, s the entropy of the normal fluid and ρs the density of the superfluid component.
To this contribution, arising from the relative motion of the two components, one must
add a dissipative contribution of the form −λ∇T . The two-fluid model is able to describe
many features of liquid helium. However, it is not completely satisfactory because the two
components cannot have independent existence; moreover, the superfluid cannot be strictly
interpreted as a Bose–Einstein condensed phase, because of intense interactions. For these
reasons, several authors have tried to describe superfluids by adding to the conventional fluid
theory a vectorial degree of freedom, to take into account the relative motion of the excitations
with respect to the bulk of the fluid (Atkin and Fox 1977, Lebon and Jou 1979) but this approach
will no longer be discussed here, as no recent developments have been proposed in this field.
Rather, we shall examine the Landau two-fluid model in the light of EIT. In the two-fluid
Extended irreversible thermodynamics revisited (1988–98)
1123
description, the corresponding Gibbs equation, formulated in the reference frame of the centre
of mass, takes the form (Lebon and Jou 1983)
d(ρs) = T −1 d(ρu) − T −1 µ dρ − (ρs /ρ)(ν n − ν s ) · d[ρn (ν n − ν s )]. (15.2)
Here, ρn stands for the density of the normal component and ρ = ρs + ρn is the total density.
The linearized evolution equations for the ideal superfluid (no viscosity) are (Mongiovı̀
1991)
∂νj
∂ρ
+ρ
=0
(mass balance)
∂t
∂xj
∂νi 1 ∂p
+
=0
(momentum balance)
∂t
ρ ∂xi
(15.3)
1 ∂qi
∂T TpT ∂νi
+
=0
(energy balance)
+
∂t
ρcv ∂xi ρcv ∂xi
∂qi λ ∂T
+
=0
(heat flux equation)
∂t
τ1 ∂xi
where pT denotes the derivative of the pressure p with respect to the temperature T . We
have assumed that the relaxation time of the heat flux is very long, in such a way that qi /τ1 is
negligible. To these equations are associated two waves, whose speeds are the solutions of the
characteristic equation
(u2 − V12 )(u2 − V22 ) − W1 W2 u2 = 0
(15.4)
= pρ ;
= λ/ρcv τ1 ; W1 = pT /ρ; W2 = T pT /ρcv . In helium II, pT is very
with
low; therefore, if one neglects W1 and W2 , one finds as solutions of (15.4) u = ±V1 , which
corresponds to sound waves where only density and velocity vibrate (first sound), and u = ±V2 ,
corresponding to waves where only temperature and heat flux vibrate (second sound).
In the presence of viscous effects, one should complete the set (15.3) with equations (2.37)–
(2.39) for the bulk viscous pressure and the deviatoric viscous pressure with τ0 ≈ 0 and τ2 ≈ 0,
because experimentally these relaxation times are very small. Now, equations (15.3) take the
more general form (Mongiovı̀ 1991, Mongiovı̀ and Romeo 1995)
∂T
∂νi
∂p
− pT
+ ρpρ
=0
(15.5a)
∂t
∂t
∂xi
∂qj
∂qhj
∂νi 1 ∂p
ς0 ∂ ∂νj
2η0 ∂ ∂νhj
−
− β 0T
− β 00 T
+
−
=0
(15.5b)
∂t
ρ ∂xi
ρ ∂xi ∂xj
∂xj
ρ ∂xj ∂xii
∂xii
∂T TpT ∂νi
1 ∂qi
+
=0
(15.5c)
+
∂t
ρcv ∂xi ρcv ∂xi
∂qj
∂qhj
λ ∂ ∂νj
λ ∂ ∂νhj
∂qi λ ∂T
+
+ ςβ 0 T 2
− β 0T
− β 00 T
+ 2ηβ 00 T 2
∂t
τ ∂xi
τ1 ∂xi ∂xj
∂xj
τ1 ∂xj ∂xii
∂xii
1
= − qi .
(15.5d)
τ1
These equations lead to the following dispersion equation for plane waves (Mongiovı̀ 1993a, b):
∂qj
4
∂qi λ ∂T
ωk 2
λ ∂ ∂νj
+ ςβ 0 T 2
+
ς+ η
− β 0T
ω2 − k 2 (V12 − W1 W2 ) + i
∂t
τ ∂xi
τ1 ∂xi ∂xj
∂xj
ρ
3
ω
4
λ
× ω2 + i − k 2 V22 + ik 2 ωT 3
ςβ 02 + ηβ 002
τ1
τ1
3
2
2
4 00
4 00
k W2 λ
k W2
2 T
0
2
2 λ
0
− ik ω
ςβ + ηβ
+ ik ωT
ςβ + ηβ
+
ρT
ρ
3
τ1
τ1
3
(15.6)
V12
V22
1124
D Jou et al
where k = kr + iks is the complex wavenumber and ω the (real) frequency. If one neglects
thermal expansion (W1 = W2 = 0), assumes that the dissipation is small (kr ks ) and that
ω is high, one obtains from (15.6) the following solutions (Mongiovı̀ 1993):
ω2
4
η
ks(1) =
ς
+
u21 = V12 ;
3
2ρu31
(15.7)
2 3 1
ω
4 002
T
λ
2
2
(2)
02
ηβ
ks =
+
+
ςβ
.
u2 = V2 ;
2u2 τ1
3
2u32 τ1
The expression for the attenuation coefficient ks(1) of the first sound is identical to that derived
by Landau and Khalatnikov in the classical two-fluid model, but the expression of ks(2) differs
in both models, since the term in (2u2 τ1 )−1 appearing in (15.7) is lacking in the Landau–
Khalatnikov model. Attenuation measurements indicate that τ1 is rather high, so that in practice
the difference for ks(2) in both models is small. A detailed comparison of the dissipative Landau–
Khalatnikov and EIT models has been carried out by Mongiovı̀ (1993, 1995); she showed that
the results of both models may be identified in the limit when 1/τ1 = 0 (but λ/τ1 finite) and
if β 0 = −(ρs T 2 )−1 . Therefore, EIT has the advantage of giving an expression of ks(2) valid for
any value of τ1 , not necessarily infinite.
To take into account the coupling between momentum and heat flux, one must consider
a relation of the form (Mongiovı̀ 1992, 1993b, Lebon and Jou 1979) P = a qq which should
be added to the linear terms in the fourth term of the left-hand side of (15.5b). This relation
has been used to describe well known experiments showing the deflection of a hanging plate
under the action of a heat flow.
16. Nuclear collisions
An important problem in nuclear physics is the establishment of the equations of state for
nuclear matter. At high energies, one of the main questions is related to the transition from
nuclear matter to quark-gluon plasma (Schukaft 1993, Mornas and Ornik 1995). The duration
of the collisions between nuclei is only one order of magnitude longer than the mean collision
time between nucleons inside the nuclei. It follows that in most collisions, the nuclei are not
thermalized. Therefore, it is logical to ask about the importance of non-equilibrium corrections
and, at higher energies, about their role in the shift of the transition line. The non-equilibrium
equations of state of EIT provide a natural frame for studying such problems.
16.1. Generalized Gibbs equation for nuclear matter
Before writing the Gibbs equation, we need expressions for the transport coefficients and the
relaxation times. They may be derived, for instance, from the kinetic theory by expanding
the Uhlenbeck–Uehling equation, which is the quantum version of the Boltzmann equation.
Danielewicz (1984) has obtained the following expressions for the shear viscosity η and the
thermal conductivity λ:
2
0.7
22
5.8(kB T )1/2
n
n
1700
+
+
(16.1)
η=
(kB T )2 n0
1 + 10−3 (kB T )2 n0
1 + 160(kB T )−2
0.4
0.15 n 1.4
0.02
0.225(kB T )1/2
n
λ=
+
+
(16.2)
−6
4
kB T n0
1 + 10 (kB T ) /7 n0
1 + 160(kB T )−2
where kB T is expressed in MeV, η in MeV fm−2 c−1 , λ in c fm−2 , the nucleon number n in
fm−3 and where n0 = 0.145 fm−3 (here, fm stands for fermi, with 1 fm = 10−15 m and c the
speed of light).
Extended irreversible thermodynamics revisited (1988–98)
1125
The corresponding relaxation times for the viscous pressure tensor P ν and the heat flux
q are (Danielewicz 1984)
1/3 h
38(kB T )−1/2 n0 850
n0 i
n
+
(16.3)
T
1
+
0.04k
τ2 =
B
(kB T )2
n0
n
1 + 160(kB T )−2 n
0.4 h
310
n0 i
57(kB T )−1/2 n0 n
τ1 =
+
(16.4)
1 + 0.1kB T
2
(kB T )
n0
n
1 + 160(kB T )−2 n
where τi is expressed in fm c−1 . The last terms in the right-hand side correspond to binary
collisions amongst nucleons, whereas the first terms describe collisions of nucleons with the
surface of the nuclei. For kB T > 6 MeV, binary collisions are more frequent whereas at
lower temperatures, a ballistic regime with collisions against the boundaries of the nucleus is
observed.
The generalized non-equilibrium entropy up to the second-order contributions in the fluxes
is given by the now classical expression
τ1
τ2
q·q−
P ν : P ν.
(16.5)
s(u, v, q , P ν ) = seq (u, v) −
2λρT 2
4ηρT
We may combine (16.1)–(16.4) with (16.5) to obtain an explicit expression for the
non-equilibrium entropy. Since the general expression would be rather cumbersome, we
particularize the results for high temperature, which is the regime of interest in the analysis of
high-energy collisions. In this case,
1.26 3 n0
1.64 n0 ν
10 2 q · q −
P : P ν.
(16.6a)
mT 3
n
mT 2 n2
ν
, the generalized entropy may be written as
For a system submitted to a viscous pressure P12
s = seq −
3.28 n0 ν 2
(P )
(16.6b)
mT 2 n2 12
ν
ν
P12
, and where the
where we have taken into account that in a shear flow P ν : P ν = 2P12
2
terms in q have been neglected.
The caloric equation of state for the temperature, i.e.
∂s
(16.7)
θ −1 =
∂u v,vP12ν
s = seq −
may be calculated explicitly if we take into account that du = c0 dT , with c0 the specific heat
per unit mass, which in this limit is c0 = 23 m−1 (recall that we set the Boltzmann constant equal
to unit). The final expression for the non-equilibrium temperature θ is (Bidar and Jou 1998)
h
i
n0
ν 2
) .
(16.8)
θ −1 = T −1 1 + 4.36 2 2 (P12
n T
We may assess the relative importance of these corrections for the collision Au + Au analysed
by Fai and Danielewicz (1996). From the results of recent simulations for the collision Au
+ Au at 400 MeV per nucleon, one has the following values for the parameters involved:
γ̇ (shear rate) ≈ 0.07 c fm−1 , ρ ≈ 0.30 fm−3 , T ≈ 44 MeV. The shear viscosity at these values
of ρ and T is η ≈ 55 MeV fm−2 c−1 . It turns out that under these conditions θ −1 ≈ 1.065T −1 ,
so that the relative non-equilibrium contribution to the temperature is of the order of 6.5%.
Furthermore, it is an easy task to compute the non-equilibrium corrections to the pressure
from the definition
∂s
π
=
.
(16.9)
θ
∂v u,vP12ν
1126
D Jou et al
ν
ν
rather than P12
Note that the quantity to be held constant during the differentiation is vP12
ν
itself, because of the non-extensive property of P12 . Expression (16.9) yields π/θ = p/T .
Therefore, from (16.8) and (16.9) we finally obtain
h
i−1
θ
n0
ν 2
π = p = p 1 + 4.36 2 2 (P12
)
.
(16.10)
T
n T
16.2. Non-equilibrium corrections to the nuclear compressibility
One of the most relevant parameters in the equations of state for nuclear matter is the so-called
isothermal nuclear-matter compressibility, which is defined in nuclear physics as
∂p
K=9
.
(16.11)
∂ρ T
The experimental results exhibit an impressive wide dispersion, even for the same nucleus.
Thus, typical values of K range from 165–220 MeV according to the kind of experiments in
which they have been obtained: some of these experiments are rather close to equilibrium
(as giant-monopole resonance) whereas other ones are obtained from nuclear collisions far
from equilibrium. To estimate the relative importance of non-equilibrium corrections in the
evaluation of K observed in nuclear collisions, Fai and Danielewicz (1996) proposed using in
K the non-equilibrium pressure π instead of the local-equilibrium pressure. If this is done,
combination of (16.10) with (16.11) yields
5.89
K = Keq +
(P ν )2 .
(16.12)
mn2 T 12
ν
Using the numerical values for the reaction Au + Au and taking into account P12
= ηγ̇ , one
finds from (16.12) that K − Keq ≈ 20 MeV. Thus, this correction may account for at least a
substantial part of the dispersion in the observed values of K.
The reader must be warned that the uncertainties in the nuclear properties such as viscosity
and relaxation time are rather high, and that the details of the flows produced during the
collisions are also uncertain. Nevertheless, the fact that the order of magnitude of the nonequilibrium corrections is close to the observed discrepancies certainly encourages a deeper
exploration of this application.
In another context, Olson and Hiscock (1990b) have studied the restrictions placed by
the stability conditions derived from EIT on the possible forms of the equations of state of
nuclear matter. In the classical approach, it is required that the sound speed is less than the
speed of light and this provides a first restriction on such equations, but EIT brings additional
limitations by requiring that the speeds of heat and viscous waves are also lower than the speed
of light.
Although we have used a non-relativistic formulation of EIT, a more detailed analysis
of high-energy collisions would certainly require a covariant formulation. Indeed, several
covariant relativistic forms of EIT have been proposed (see Jou et al (1998a), Pavón (1992),
and Gariel and Le Denmat (1994) for bibliographical reviews). The use of the generalized
transport equations of EIT allows one to avoid not only causality problems related to speeds
higher than the speed of light, but also severe instability problems (Olson 1990, Hiscock and
Lindblom 1988a, b, Hiscock and Olson 1989, Ruggeri 1989, Geroch and Lindblom 1990,
1991, Lindblom 1996).
17. Viscous cosmological models
Thermodynamics plays a key role in cosmological problems through the energy–momentum
tensor Tµν , which appears on the right-hand side of the Einstein equations of general relativity
Extended irreversible thermodynamics revisited (1988–98)
1127
(see, for instance, Wald 1984)
Rµν − 21 gµν R = 8πGTµν − 3gµν
(17.1)
where Rµν is the Ricci tensor associated to the metric tensor gµν , R the scalar curvature, G
the Newtonian gravitational constant and 3 the cosmological constant.
These equations are very complicated, but they may be simplified by assuming isotropy
and homogeneity at large scales (the so-called cosmological principle), which leads to a
Friedmann–Robertson–Walker (FRW) metric of the form
ds 2 = −c2 dt 2 + R 2 (t)[(1 − κr 2 )−1 dr 2 + r 2 (dθ 2 + sin2 θ dφ 2 )]
(17.2)
with s the spacetime separation between two arbitrary events; r, θ , and φ the comoving
coordinates and R(t) the cosmic scale factor. The parameter κ is a constant related to the
spatial curvature which may take only three values: κ = 0 (flat space sections), κ = ±1
(curved spatial sections). Here we will only consider the case κ = 0.
To calculate R(t) from the Einstein equations, one needs the energy-momentum tensor.
Its most general form compatible with homogeneity and isotropy is


−ε
0
0
0
ν
0
0 
 0 p+p
(17.3)
Tµν = 

0
0
p + pν
0
ν
0
0
0
p+p
with ε the internal energy density, p the pressure and p ν the bulk viscous pressure. Unlike
the perfect fluid hypothesis, which states that pν = 0, it is assumed here that the bulk viscous
pressure does not necessarily vanish.
Indeed, several phenomena can generate viscous dissipation on cosmological scales
(Barrow 1988, Lima et al 1988, Oliveira and Salim 1988, Pavon et al 1991, Maartens 1995 and
references therein). Neutrinos (whether massless or massive) interacting with matter represent
an efficient mechanism of dissipation owing to their long mean-free path; likewise, photons
in contact with matter, e.g. electrons, constitute a radiative fluid able to generate entropy.
As other examples, let us mention vacuum polarization and particle production; interactions
between matter and radiation, quarks and gluons, different components of dark matter, decay
of massive superstring modes, gravitational string production, and grand unification phase
transitions (Pavón and Zimdahl 1993, 1994, Zimdahl 1996, Zimdahl et al 1996, Chimento and
Jakubi 1997).
Now the non-trivial component of the Einstein equations take the simple form
2
3
8πG
Ṙ
ε+
(17.4a)
=
R
3
3
4πG
R̈
=−
[ε + 3(p + p ν )]
(17.4b)
R
3
where an upper dot stands for the time derivative. The set (17.4) will be closed by introducing
an equation of state for p and an evolution equation for pν , for instance
p = (γ0 − 1)ε,
τ0 ṗν + p ν = −3ζ H.
(17.5)
The first of these equations is standard; in the second one, which is typical of EIT, use was
made of ∇ · ν = 3H , which follows from the result ν = dr /dt = H r , with r the position
vector of the particles, and H = Ṙ/R the Hubble factor. The coefficient γ0 is supposed to
satisfy 1 6 γ0 6 2. The value γ0 = 43 corresponds to radiation (p = ε/3) and γ0 = 1 to an
ideal non-relativistic gas (kB T mc2 ). The bulk viscosity ζ and the relaxation time τ0 are
assumed to depend on ε through (Belinskii et al 1979)
ζ = αε ν ,
τ = ζ /ε = αεν−1 .
(17.6)
1128
D Jou et al
With the above choice, the viscous signals propagate with a speed equal to c, since the speed
of propagation of viscous signals is ν = [ζ /(ετ )]1/2 .
17.1. Bulk viscosity and the evolution of the Universe
Assume that the cosmological constant is vanishing. Elimination of ε in (17.4) yields
2
R̈ 3γ0 − 2 Ṙ
+
= −4π Gpν .
(17.7)
R
2
R
In the standard cosmological model, viscous effects are ignored (pν = 0), and one is led to
R ∼ t 1/2 and R ∼ t 2/3 for radiation- and matter-dominated universes, respectively (Wald
1984).
Introduction of bulk viscosity may have several consequences, depending on the relation
between ζ and ε. If ζ is proportional to ε, the big-bang singularity with infinite spacetime
curvature would not occur (Murphy 1973). However, more realistic models are based on (17.6),
in which case the big-bang singularity still persists for ν 6= 1 (Oliveira and Salim 1988).
In most models, bulk viscosity is introduced by means of classical transport equations, as
in the second equation of (17.5), but without the relaxation term. This is, however, inconsistent,
since it leads to infinite speeds and to unstable solutions. The consequences of the introduction
of relaxation terms are important when the product of the relaxation time and the Hubble
parameter is larger than or comparable to unity. This may happen at several stages of the
expansion of the Universe, because different components of the cosmic fluid may have different
collision times.
A simple model is obtained by combining (17.4b) (Pavón et al 1991) and the second
equation of (17.5), which may be written in terms of the Hubble parameter H as
τ Ḧ + Ḣ (3τ γ0 H + 1) + H ( 23 γ0 H − 12π Gς ) = 0.
(17.8)
In some particular situations, it is possible to obtain simple solutions. For instance, if
Ḣ /H 1, the viscoelastic effects (τ ṗν ) in (17.5b) are dominant over the purely viscous
ones (p ν ), so that the classical nonviscous behaviour R ≈ t 1/2 (radiation) and R ≈ t 2/3 (dust)
are changed to R ≈ t 0.35 (radiation) and R ≈ t 0.41 (dust) (Zakari and Jou 1993a). In general,
the solutions of (17.8) are much more complicated. First, its resolution demands that two initial
conditions are specified for H and Ḣ . In the standard theory, one single initial condition for
H is required and is usually taken to be H (0) = ∞. In the present case, the initial value of H
can be finite, even negative. This means that the analysis of the primordial Universe requires
not only a unified theory of the basic interactions, but also a thermodynamic description of the
collective effects, which are usually ignored in standard formalisms.
Inserting into equation (17.8) the relationships (17.6) for ζ and τ , and using (17.4a) leads
to
12π Gα 2n+1
3
H
γ0 −
=0
(17.9)
αβ n H 2n Ḧ + Ḣ (3αβ n γ0 H 2n+1 + 1) + H 2
2
βν
with n = ν − 1 and β a shorthand for 3/(8π G). Upon introducing the quantities
1/(2n+1)
γ0 8π G n
1
H
,
H0 =
,
ν 6=
(17.10)
h=
H0
3α
3
2
and rescaling the time as t ∗ = H0 t, (17.9) may be cast into the form
γ0 h2n ḧ + 3ḣ(γ0 h2n+1 + 1) + 29 γ0 h2 (1 − h2n+1 ) = 0.
(17.11)
∗
From now on, an upper dot will denote derivation with respect to t ; the asterisks will be
dropped whenever no confusion arises. The second-order differential equation (17.11) which
Extended irreversible thermodynamics revisited (1988–98)
1129
governs the evolution of the reduced Hubble parameter h has two different stationary trivial
solutions, h = 1 and h = 0. The first implies inflationary expansion with a constant rate given
by H0 .
Specific solutions (17.11) have been studied, among others, by Oliveira and Salim (1988)
and Novello et al (1990), and Pavon et al (1991). These authors discuss its behaviour near
the steady solutions analytically; they showed that, in contrast with the non-causal model, the
classical de Sitter solution is no longer stable in the remote past for ν > 21 , and that a deflation
is excluded by causality. Chimento and Jakubi (1993, 1997) found exact solutions for ν = 21
and examined analytically the asymptotic stability of distinct families of solutions for arbitrary
ν.
It is worth stressing that the second equation of (17.5) for pν is not the most general
evolution equation obtained from EIT. Indeed, neglecting heat flux and shear viscous pressure
terms in the expression of the entropy four-vector the latter reads, up to the second order in
the viscous pressure, as
τ (p ν )2
µ
(17.12)
uµ .
S = seq −
2ζ T
Taking the full derivative of this expression, one obtains for the entropy production
pν
τ ν 1 ν τ µ
µ
µ
S;µ = −
u;µ + ṗ + p T
u
(17.13)
T
ζ
2
ζ T ;µ
where the semicolon stands for the covariant derivative. This suggests, instead of (17.5), the
non-truncated equation
ζ̇
Ṫ
1
τ̇
p ν + τ ṗ ν = −3ζ H − p ν τ 3H + − −
.
(17.14)
2
τ
ζ
T
The behaviour of the cosmological models depends both on the equations of state for p,
T , τ , and ζ and the transport equation being used. For instance, Hiscock and Salmonson
(1991), who used (17.14) coupled with the equations of state for the Boltzmann gas, found
no inflationary phase, in contrast to the prediction of the truncated equation of (17.5). With
simpler equations of state p ν = λρ, τ = αρ m , ζ = τρ other authors (Zakari and Jou 1993b,
Romano and Pavón 1994) have found that an inflationary phase remains possible in the nontruncated model, but with a different expansion rate than in the truncated version of (17.5).
Gariel and Le Denmat (1994) have studied the relationship between both formalisms. Pavón
et al (1996a, b) found that an exponential inflation driven by a dissipative bulk stress due
to matter creation cannot be a causal solution of the FRW cosmology, but that a power law
inflation is a plausible solution. Starting from equation (17.14), Méndez (1996) found some
exact solutions for a de Sitter expansion. Mendez and Pavón (1996b) have derived exact
solutions for the intermediate epoch connecting the radiation era to the matter era.
In view of the difficulty of obtaining analytical solutions, qualitative approaches of causal
anisotropic viscous cosmological models have been proposed by several authors, such as Coley
et al (1996) and Méndez and Triginer (1996). They carried out a detailed analysis of the stability
of the de Sitter and static cosmological models, both in the truncated and the non-truncated
versions.
17.2. Other topics: particle production, time-dependent cosmological constant,
astrophysical problems
There are other topics of interest related to the effective bulk viscosity. They concern particle
production, or the possibility of a variable cosmological constant, which in some situations
1130
D Jou et al
may be described by means of an apparent bulk viscosity. Other situations of interest are found
in astrophysics.
(a) Decay and production of particles in the Universe. Such a decay or production finds its
origin in several different effects such as, for instance, the decay of scalar particles, production
of relativistic particles in the reheating phase of inflationary eras, electron–positron annihilation
after neutrino decoupling, decay of heavy bosons to quarks and leptons, or the creation of
particles by the gravitational field. Prigogine et al (1989) were able to frame these effects
in a thermodynamic description by formulating a balance equation for the number density of
created particles in addition to the Einstein field equations. Another method is to model the
loss and source terms by an effective viscous pressure (Sudharan and Johri 1994, Sussman
1994, Triginer and Pavón 1994, Gariel and Le Denmat 1995, Abramo and Lima 1996, Pavon
et al 1996a, b). The usual FRW universe is modified in such a way that if the creation rate is
sufficiently high at Planck scales, the initial singularity may be avoided (Zimdahl and Pavón
1994a, b).
Furthermore, spacetime geometries more complicated than the homogeneous isotropic
FRW model have been studied. Let us mention Bianchi I and III universes which include
bulk pressure and shear viscous stresses, both in the truncated and the non-truncated models
(Romano and Pavón 1993, 1994), and descriptions taking into account peculiar velocities of
the galaxies with respect to the global cosmic motion (Triginer and Pavón 1995). Several of
these macroscopic results have also been confirmed from the kinetic theory of gases in an
expanding universe (Triginer et al 1996, Zimdahl et al 1996).
(b) Time-dependent cosmological ‘constant’. A possible solution to the puzzle of the
cosmological constant (i.e. to understand its low present value in contrast to its high values in
the past) has been to introduce a variable cosmological ‘constant’ in the Einstein equations.
It has been argued that because the cosmological term is proportional to the energy density
of the vacuum it should decrease with the expansion as a function of the scale factor. Some
authors assume that it decays as 3 ≈ R −2 whereas others take a decrease in t −2 compatible
with the present low value of 3. The model with 3 ≈ t −2 is equivalent to a viscous fluid
with null cosmological constant (Beesham 1993). The time variation of the cosmological
constant contributes to the entropy production, as it describes the transformation of vacuum into
particles. When the chemical potential of these particles is neglected, the entropy production
is found to be (Méndez and Pavón 1996b)
c5 3̇
9ζ H 2
α
−
.
(17.15)
=
S;α
T
8πG T
Several expressions have been proposed for the bulk viscosity ζ . Calvão et al (1992) consider
a bulk viscosity proportional to ρ while Méndez and Pavón (1996a, b) take ζ ≈ ρ 1/2 (which
turns out to be at variance with observation) and ρ 3/2 . Other analyses may be found in Arbab
and Abdel-Rahman (1996).
(c) Astrophysics: gravitational collapse. The relaxational terms in the transport laws have
been used to interpret fast explosions or implosions in astrophysics (Martı́nez and Pavón 1994,
Martı́nez 1996, Herrera and Falcon 1995a–c, Herrera et al 1997). In rapid processes, as in
the fast collapse phase preceding neutron star formation, the relaxational effects are important
and the fluid is far from hydrostatic equilibrium. This is in contrast with usual analyses,
which consider that the evolution of the star may be regarded as a sequence of hydrostatic
models. Moreover, in contrast with cosmological problems, two different metrics must be
introduced, one for the interior and the other for the exterior of the star. Furthermore, the loss
of homogeneity implies the appearance of a heat flux, which plays an important role. Among
other results, it turns out (Martı́nez and Pavón 1994) that the behaviour of the system depends
strongly on the observer: for an observer at rest at infinity, the total mass loss of the star is
Extended irreversible thermodynamics revisited (1988–98)
1131
the same but with greater speed (i.e. higher luminosity) than in the nonviscous case, whereas
the comoving observer perceives a larger mass loss. Another result is that the energy of the
neutrinos at the surface of the start is not correlated with that at the interior (Martı́nez 1996).
Prisco et al (1996) and Herrera et al (1997) have shown that when the heat flux relaxes, the
evolution of the star depends critically on a parameter defined in terms of the thermal relaxation
time, thermal conductivity, proper energy density and pressure.
Starting from a Maxwell–Cattaneo equation, Herrera and Falcon (1995a–c) have revised
the criterion for the secular stability of nuclear burning in accreting neutron star envelopes,
which is at the basis of x-ray burster models, for times shorter than the effective relaxation time
of the heat flux. The characteristic timescales for the growth of the nuclear burning instability
spans a wide range, from milliseconds to minutes. Since the relaxation time of the heat flux
is of the order of milliseconds for a highly degenerate matter, it follows that the relaxational
effects will be of interest when the instability is fast enough. These relaxations produce a fast
oscillation in the luminosity as a precursor of the x-ray burst.
(d) Flux limiters for viscous pressure. The non-truncated equation (17.14) cannot be
considered as a full nonlinear theory. Indeed, it has been obtained by assuming a linear
relation between the flux and the generalized thermodynamic force. Maartens and Méndez
(1997) have recently proposed a nonlinear generalization fo the transport law; instead of taking
pν = −ζ χ, with χ the generalized thermodynamic force conjugated to pν , these authors
suggest a nonlinear generalization of the form
pν = −
ζχ
1 + τ∗ χ
(17.16)
where τ∗ is a characteristic timescale for nonlinear effects. This equation is the simplest
nonlinear generalization leading to a saturation value for pν in the limit of very high values of
the thermodynamic force χ, in contrast to the linear law for which p ν diverges at large χ . In their
paper, Maartens and Méndez impose that the generalized entropy S = Seq − (τ/2nT ζ )(pν )2
is always positive. This choice allows for inflationary behaviour. Other more restrictive
conditions may be introduced, which could make impossible inflation in cosmology, and could
also modify some of the astrophysical results concerning fast collapses of stars.
18. General conclusions
The main questions to be addressed are the following: to what extent does EIT contribute to
a better understanding of thermodynamics of irreversible processes? What kind of problems
are presently better apprehended by applying the methods of EIT? We have tried to answer the
first question in sections A and B while the second question was addressed in section C.
First of all, EIT clearly emphasizes the strong correlation between the transport equations
and thermodynamics. One cannot formulate transport equations independently of their
thermodynamic background. For instance, it follows from the present work that introducing
relaxation terms in the transport equations results in a modification of the expression of entropy
whereas the presence of non-local terms leads to a generalization of the expression of the
entropy flux. In that respect, EIT suggests that in the future more attention should be dedicated
to this unifying aspect, which is usually forgotten, because of the almost exclusive focus on
transport equations for themselves.
From a fundamental point of view, EIT has raised more challenging problems during
the last decade than in the initial stage of its formulation. It is our opinion that, rather
than invalidating the basic concepts of the theory, this has raised important and fundamental
questions in non-equilibrium thermodynamics and statistical mechanics. Among the open
1132
D Jou et al
questions are the meaning of temperature and entropy in non-equilibrium states, and the relation
between the H theorem and the formulation of the second law in non-equilibrium processes
(Lebon et al 1992, 1996, Garcı́a-Colı́n 1995). Another important problem is the formulation of
non-equilibrium equations of state and, in particular, the definitions of temperature and pressure
outside equilibrium. Definite answers are undoubtedly beyond the scope of EIT, but the latter
may be helpful in proposing some ways out and in formulating them in explicit terms. It should
also be recalled that in classical irreversible thermodynamics such fundamental questions do
not arise as they are eluded by appealing to the local-equilibrium postulate.
Concerning the selection of variables, it is shown in section A that the use of the
‘equilibrium’ slow variables complemented by fast variables taking the form of fluxes of
mass, momentum and energy is generally sufficient to solve a great variety of problems. The
time-evolution equations of these variables generalize the classical transport equations of
Fourier, Fick, Newton: they display memory effects and may contain nonlinear and non-local
contributions; they guarantee, in addition, that the causality principle demanding that effect
comes after the cause is satisfied. Nevertheless, we are faced with a dilemma. It is evident that
an increasing number of variables is necessary to describe high-frequency responses of systems:
Weiss and Müller (1995) have computed the density correlation function by including in it more
and more higher-order fluxes, and have compared them with the experimental results obtained
from light scattering in gases. They show that after taking 230 moments, the results are no
longer affected by the addition of more moments. Thus, depending on the details that one wants
to reproduce, it appears that one should include an increasing number of variables. On the other
hand, the problem soon becomes intractable when too large a number of variables are involved.
In this context, let us mention the work of Dedeurwaerdere et al (1996) which, starting from
an infinite number of fluxes, introduces a method based on an asymptotic continued-fraction
expansion of the frequency- and wavelength-dependent transport coefficients. This method
allows one to keep only a small number of independent variables and to include the effects of
all the others in some effective relaxation times. This topic has also been examined from the
microscopic point of view (Ichiyanagi 1995b, 1996, Luzzi et al 1998). Instead of the common
strategy, which consists of eliminating all the fast variables, in order to describe the most
prominent features of high-frequency response, they include a few relevant fast variables and
eliminate the others by a procedure similar to the renormalization scheme used in the analysis
of critical phenomena.
Another difference between EIT and the classical theories is the formulation of nonequilibrium equations of state: as a matter of fact, the latter are now flux dependent. A
typical example is the caloric equation of state wherein the internal energy does not only
depend on the local-equilibrium temperature but, in addition, on the fluxes. This dependence
is supported by experimental results on the decay of photoexcited plasma in semiconductors in
the presence of an electric current, or by computer simulations. Moreover, such a generalized
equation of state is a consequence of the existence of a non-equilibrium temperature different
from the classical local-equilibrium temperature introduced in most formalisms. Kinetic
theory developments indicate that the non-equilibrium temperature is directly related to the
translational kinetic energy of the particles in the plane normal to the heat flux. More generally,
it is also expected that different degrees of freedom may have different ‘temperatures’, in
such a way that thermometers sensible to different degrees of freedom will indicate different
temperatures. Similar conclusions hold for the non-equilibrium pressure and non-equilibrium
chemical potential which are also allowed to be flux dependent and whose properties and
consequences are discussed in length in the present review (sections 3, 11 and 16).
The relation between the Boltzmann and the macroscopic entropy was clarified in
section 7, where we also discussed the link between the H theorem and the second law of
Extended irreversible thermodynamics revisited (1988–98)
1133
thermodynamics. Indeed, the H theorem refers to the evolution of the microscopic Boltzmann
entropy, defined in terms of the one-particle distribution function which is the exact solution
of the Boltzmann equation, whereas the second law of thermodynamics is formulated in terms
of a macroscopic entropy, defined from a reduced number of macroscopic variables.
The debate about such fundamental questions as the definitions of non-equilibrium
temperature, pressure, entropy and the H theorem has been given a new impetus mainly thanks
to the recent development of EIT. Thus, it was shown that the latter can be described in terms of
a generalized Hamiltonian structure in the light of formulation of GENERIC (general equations
of non-equilibrium reversible and irreversible coupling) developed recently by Grmela et al
(1998). A simple illustration is provided in section 5.
It is well recognized that a macroscopic theory is not fully self-consistent as it contains
unknown phenomenological coefficients which cannot be determined and interpreted without
referring to experiments or microscopic model theories. In this review, we have frequently
appealed to the kinetic theory of gases and the information theory to settle the foundations and
results of EIT.
In writing this review, our aim was also to provide a list of applications which are more
easily treated by applying the methods of EIT. This is mainly the case when the relevant
equations describing a physical or chemical process are of the hyperbolic or telegrapher type.
As a typical application, let us mention heat transport at very low temperature in dielectric
crystals.
Second-sound propagation was evidenced experimentally during the 1960s and explained
theoretically within the framework of EIT. Second sound was first discovered in liquid helium II,
and the peculiar properties of He are usually investigated by means of Landau’s classical twofluid model. The problem has been revisited recently by several authors who proposed a more
general description based on EIT (see section 15).
Radiation hydrodynamics is another example of non-classical heat transport. This is
not surprising as radiation has a well-defined maximum speed, and therefore the need of a
hyperbolic equation for the description of energy transport is widely recognized. A particularly
interesting result arising from EIT is that it imposes that a signal such as the heat flux, cannot
reach an unbounded value but is ‘saturated’ by a limiting value equal to the energy density
times a maximum speed, say the speed of light in radiative heat transfer. This saturation
effect cannot be described by the classical Fourier law. The use of flux limiters, described in
section 14.2, is of capital importance in astrophysics.
Another privileged domain of application of EIT is that of polymer solutions, as they
exhibit rather long relaxation times for the viscous pressure tensor. EIT has proved to be of
use in obtaining the relevant constitutive equations and, in that respect, it confirms in several
cases the equations provided by other approaches. In contrast to the formalisms based on
internal variables, EIT identifies from the start the viscous pressure as an additional variable.
This allows one to unify the description of very large classes of material systems, passing
from ideal gases to viscoelastic materials, the fundamental differences being the value of the
relaxation time and the microscopic form of the pressure tensor. Another more exclusive
advantage of EIT is that it predicts generalized equations of state which are particularly well
suited to study phase transitions in polymer solutions, as shown in section 11.3. It is well
known that two effects, a pure thermodynamical one and a dynamical one, may contribute to
such phase changes. Since both effects are naturally incorporated in the description of EIT, it
is not surprising that one obtains a better agreement with experiments concerning the shift of
the critical temperature or the spinodal curve than with more classical approaches. In parallel
with non-Fourier heat transport, non-Fickian mass diffusion finds a natural place in EIT. Nonstandard diffusion is of practical importance mainly in polymer diffusion, like sorption or
1134
D Jou et al
permeation of a solvent in a film. As shown in section 12.2, EIT is also well suited for the
description of Taylor’s diffusion which plays an important role in many practical applications.
Another problem of actual interest in industry is the transport of electrical carriers in
nanoscale microelectronic devices. The usual Boltzmann equation is too intricate to deal with,
and a description based on a few moments of the distribution function is clearly advantageous,
because these are precisely the moments related with the observable variables. The crucial
problem is to operate the truncation of the hierarchy of equations for the moments obtained from
the Boltzmann equation. EIT provides thermodynamic restrictions of the possible truncations,
and may therefore be considered as a valuable basis for a hydrodynamical description of
transport in microelectronic devices.
Other topics have recently been analysed within the framework of EIT. Let us mention some
astrophysical phenomena, the developments of cosmological models including dissipative
effects, mainly bulk viscosity, and the role of non-equilibrium contributions in the equations
of state of nuclear matter. More recently EIT has been applied to new subjects like
climatic modelling (Sancho and Llebot 1994b), turbulence (Sancho and Llebot 1994a, b),
and hydrodynamic instabilities (Franchi and Straughan 1994a, b, Lebon et al 1994).
The variety of subjects covered by the present review demonstrates that the domain of
applications of EIT has been considerably enlarged during the last decade. EIT has been shown
to be the specific corpus wherein a wide variety of problems find a natural accomodation. A
more thorough treatment of these topics is a typical challenge of EIT at the level of applications.
In summary, EIT offers new perspectives which must be carefully scrutinized, and
henceforth enlarges the domain of applications of classical thermodynamics. In this review,
we have excerpted the main lines of research, with special emphasis on topics which are linked
to our personal preferences and knowledge of the field. Other authors would probably have
underlined other topics but we think that a diversity of points of view is always healthy to
foster the interest in a field of research. The present time is rather exciting in relation with the
development of EIT. The fundamental questions raised above are not particular to EIT, but they
are rather general and fundamental in non-equilibrium thermodynamics and statistical physics.
Comparison between EIT and other approaches should be encouraged. It is also hoped that in
the future more applications with industrial impact will be developed.
Acknowledgments
In dealing with the diversity of topics discussed in this review we have benefited from many
discussions with different researchers: A M Anile (transport in semiconductors), J Camacho
(Taylor dispersion), M Criado-Sancho (polymer solutions under shear), R Domı́nguezCascante (radiation hydrodynamics), M Grmela (Hamiltonian structure), R Luzzi (information
theory), and D Pavón (cosmology), and the referees for their useful suggestions and careful
analysis of the first version of this paper.
This work has been supported by a EU grant in the framework of the Programme of
Human Capital and Mobility (European Thermodynamic Network ERB-CHR-XCT 920 007).
DJ and JC-V have also been supported by the Dirección General de Investigación Cientı́fica
y Técnica of the Spanish Ministry of Education and Culture under grant PB94-0718. GL
was supported by Interuniversity Pole of Attraction (IUPA, convention IV 06) initiated by
the Belgian State, Science Policy Programming. The website http://circe.uab.es/eit with the
bibliographical database on EIT and related topics has been supported by the Direcció General
de Recerca of the Generalitat of Catalonia under grant 1997SGR00387.
Extended irreversible thermodynamics revisited (1988–98)
1135
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