### 5.5 Specific results for a non-degenerate relativistic gas

For a relativistic gas Jüttner [22, 23] 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 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
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 ’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 ’s can be calculated from the thermal equation of state
and the ’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 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].