A pioneer of this strategy was Carl Eckart [14, 15, 16] who - as early as 1940 - established the thermodynamics of irreversible processes, a theory now universally known by the acronym TIP. The third of Eckart's three papers addresses the relativistic theory of a fluid. Eckart's theory is an important step away from equilibria toward non-equilibrium processes. It provides the Navier-Stokes equations for the deviatoric stress and a generalization of Fourier's law of heat conduction. The latter permits a heat flux to be generated by an acceleration, or a temperature gradient to be equilibrated by a gravitational field.
But Eckart's theories - the relativistic and non-relativistic ones - have one draw-back: They lead to parabolic equations for the temperature and velocity and thus predict infinite pulse speeds. Naturally relativists, who know that no speed can exceed c, are particularly disturbed by this result and they like to call it a paradox.
Cattaneo [7] proposed a solution of the paradox as far as it concerns heat
conduction
. He reasoned that under rapid changes of temperature the heat
flux is somewhat influenced by the history of the temperature
gradient and he was thus able to produce a hyperbolic equation
for the temperature - actually a telegraph equation.
Müller [35
,
37] incorporated this idea into TIP and came up with a fully
hyperbolic system for temperature
and
velocity. He calculated the pulse speeds and found them to be of
the order of magnitude of the speed of sound, far removed from
c
. And indeed, neither Cattaneo's nor Müller's arguments have
anything to do with relativity, although Müller [35] also formulated his theory relativistically. The theory became
known as Extended Thermodynamics, because the canonical list of
fields - density, velocity, temperature - is extended in this
theory to include stress and heat flux, 14 fields altogether.
The pulse speed problem may not be the most important question
in thermodynamics but it is a question that can be answered, and
has to be answered, and so there was a series of papers on the
problem all using extended thermodynamics of 14 fields.
Israel [21] - who reinvented extended thermodynamics in 1976 - and
Kranys [24] and Stewart [46], and Boillat [2], and Seccia & Strumia [44] all calculate the pulse speed for classical as well as for
relativistic gases, degenerate and non-degenerate, for Bosons and
Fermions, and for the ultra-relativistic case. Actually in some
of these gases the pulse speed reaches the order of magnitude of
c
but it never exceeds it.
So far, so good! But now consider this: The 14 fields
mentioned above are the first
moments
in the kinetic theory of gases and the kinetic theory knows many
more moments. In fact, in the kinetic theory we may define
infinitely many moments of an increasing tensorial rank. And so
Müller and his co-workers, particularly Kremer [25,
26], Weiss [49,
51,
50] and Struchtrup [47], came to realize that the original extended thermodynamics was
not extended far enough. Guided by the kinetic theory of gases
they formulated many-moment theories. These theories have proved
their validity and relevance for quickly changing processes and
processes with steep gradients, in particular for light
scattering, sound dispersion, shock wave structure and radiation
thermodynamics. And
each theory predicts a new pulse speed
. Weiss [49
], working with the non-relativistic kinetic theory of gases,
demonstrated
that the pulse speed increases with an increasing number of
moments.
Boillat & Ruggeri [5]
proved
this observation and - very recently - Boillat &
Ruggeri [6
] also proved that the pulse speed tends to infinity in the
non-relativistic kinetic theory as the number of moments becomes
infinite. As yet unpublished is the corresponding result by
Boillat & Ruggeri [3
,
4
] in the relativistic case by which the pulse speed tends to
c
as the number of moments increases. These results put an end to
the long-standing paradox of pulse speeds - 50 years after
Cattaneo; they are reviewed in Chapters
3
and
4
.
The quest for macroscopic field theories with finite pulse speeds has proved heuristically useful for the discovery of the formal structure of thermodynamics, relativistic and otherwise. This structure implies
The latter property is essential for finite speeds
and
for the well-posedness of initial value problems which is a
feature at least as desirable as finite speeds. The formal
structure of the theory is described in Chapter
2
; it was constructed by Ruggeri and his co-workers, particularly
Strumia and Boillat, see [43,
41,
5]. A convenient presentation may be found in the book by Müller
& Ruggeri [39
] of which a second edition has just appeared [40
].
Chapter
5
presents extended thermodynamics of viscous, heat-conducting
gases due to Liu, Müller & Ruggeri [31], a theory of 14 fields. That chapter demonstrates the
restrictive character of the thermodynamic constitutive theory by
showing that most constitutive coefficients can be reduced to the
thermal equation of state. Also new insight is provided into the
form of the transport coefficients: bulk- and shear-viscosity,
and thermal conductivity, which are all explicitly related here
to the relaxation times of the gas.
This whole review is concerned with a macroscopic theory: Extended thermodynamics. It is true that some of the tenets of extended thermodynamics are strongly motivated by the kinetic theory of gases, for instance the choice of moments as variables. But even so, extended thermodynamics is a field theory in its own right, it is not kinetic theory.
The kinetic theory, complete with Boltzmann equation and Stoßzahlansatz, offers another possibility of discussing finite propagation speeds - or speeds smaller than c in the relativistic case. Such discussions are more directly based on the observation that the atoms cannot be faster than c . Thus Cercignani [8] has directly linked the phase speed of small harmonic waves to the speed of particles and proved that the phase speeds are smaller than c . Cercignani & Maiorana in a follow-up paper [9] have exploited the full dispersion relation to calculate phase speeds and attenuation as functions of frequency, albeit for a simplified collision term. Earlier works on the kinetic theory which address the question of propagation speeds include Sirovich & Thurber [45] and Wang Chang & Uhlenbeck [48]. These works, however, are not subjects of this review.
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Speeds of Propagation in Classical and Relativistic
Extended Thermodynamics
Ingo Müller http://www.livingreviews.org/lrr-1999-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |