in all events
. Both
and
are Lorentz tensors. The energy-momentum tensor is assumed
symmetric so that it has 10 independent components.
For the determination of these fields we need field equations and these are formed by the conservation laws of particle number and energy-momentum, viz.
and by the equations of balance of fluxes
is the flux tensor - it is completely symmetric -, and
is its production density. We assume
so that among the 15 equations (79,
80
,
81
) there are 14 independent ones, which is the appropriate number
for 14 fields.
The components of
and
have the following interpretations
The motivation for the choice of equations (79,
80
,
81
), and in particular (81
), stems from the kinetic theory of gases. Indeed
and
are the first two moments in the kinetic theory and
and
are the first two equations of transfer. Therefore it seems
reasonable to take further equations from the equation of
transfer for the third moment
and these have the form (81
). In the kinetic theory the two conditions (82
) are satisfied.
The set of equations (79,
80
,
81
) must be supplemented by constitutive equations for the flux
tensor
and the flux production
. The generic form of these relations in a viscous,
heat-conducting gas reads
If the constitutive functions
and
are known, we may eliminate
and
between (79
,
80
,
81
) and (84
) and obtain a set of field equations for
,
. Each solution is called a
thermodynamic process
.
It is clear upon reflection that this theory, based on (79,
80
,
81
) and (84
), provides a special case of the generic structure explained in
Chapter
2
.
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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 |