] has derived the phase density for Bosons and Fermions, namely
The latter equation is valid in the rest frame of the gas.
is the atomic four-momentum and we have
. Jüttner has used these phase densities to calculate the
equations of state. For the non-degenerate gas he found that
Bessel functions of the second kind, viz.
are the relevant special functions. The thermal equation of
state
reads
where 1 /
y
is the smallest phase space element. From (102
) we obtain with
and
and hence
The transport coefficients read
It is instructive to calculate the leading terms of the
transport coefficients in the non-relativistic case
. We obtain
It follows that the bulk viscosity does not appear in a non-relativistic gas. Recall that the coefficients 1/ B are relaxation times of the order of magnitude of the mean-time of free flight; so they are not in any way ''relativistically small''.
Note that
,
and
are measurable, at least in principle, so that the
B
's may be calculated from (105). Therefore it follows that the constitutive theory has led to
specific results. All constitutive coefficients are now explicit:
The
C
's can be calculated from the thermal equation of state
and the
B
's may be measured.
It might seem from (106) and (97) that the dynamic pressure is of order
but this is not so as was recently discovered by Kremer &
Müller [27]. Indeed, the second step in the Maxwell iteration for
provides a term that is of order
, see also [28]. That term is proportional to the second gradient of the
temperature
T
so that it may be said to be due to heating or cooling.
Specific results of the type (104,
105) can also be calculated for degenerate gases with the thermal
equation of state
for such gases. That equation was also derived by Jüttner [23]. The results for 14 fields may be found in Müller &
Ruggeri [39,
40].
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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 |
