follows from (19
) and (33). We obtain
On the other hand statistical mechanics defines the four-flux of entropy by (e.g. see Huang [20])
k is the Boltzmann constant and 1/ y is the smallest phase space element.
Comparison shows that we must have
and hence, by differentiation with respect to
,
so that
f
is the phase density appropriate to a degenerate gas in
non-equilibrium. Differentiation of (39) with respect to
proves the inequality (35).
Therefore symmetric hyperbolicity of the system (34) and hence the concavity of the entropy density with respect to
the variables
is implied by the moment character of the fields and the form of
the four-flux of entropy.
For a non-degenerate gas the term
in the denominator of (38) may be neglected. In that case we have
hence
and therefore the field equations (23), (34) assume the form
Note that the matrices of coefficients are composed of moments in this case of a non-degenerate gas.
We know that a non-degenerate gas at rest in equilibrium exhibits the Maxwellian phase density
n
and
T
denote the number density and the temperature of the gas in
equilibrium. Comparison of (43) with (40) shows that only two Lagrange multipliers are non-zero in
equilibrium, viz.
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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 |
