5.6 Characteristic Speeds in a 5 Relativistic Thermodynamics of Gases. 5.4 The Laws of Navier-Stokes

5.5 Specific Results for a Non-Degenerate Relativistic Gas 

For a relativistic gas Jüttner [22, 23Jump To The Next Citation Point In The Article] has derived the phase density for Bosons and Fermions, namely

  equation1652

The latter equation is valid in the rest frame of the gas. tex2html_wrap_inline4173 is the atomic four-momentum and we have tex2html_wrap_inline4439 . 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.

  equation1665

are the relevant special functions. The thermal equation of state tex2html_wrap_inline4351 reads

  equation1672

where 1 / y is the smallest phase space element. From (102Popup Equation) we obtain with tex2html_wrap_inline4445 and tex2html_wrap_inline4447

  equation1686

and hence

  equation1692

The transport coefficients read

  equation1718

It is instructive to calculate the leading terms of the transport coefficients in the non-relativistic case tex2html_wrap_inline4449 . We obtain

  equation1738

  equation1743

  equation1747

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 tex2html_wrap_inline4425, tex2html_wrap_inline4427 and tex2html_wrap_inline4429 are measurable, at least in principle, so that the B 's may be calculated from (105Popup Equation). 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 tex2html_wrap_inline4351 and the B 's may be measured.

It might seem from (106Popup Equation) and (97Popup Equation) that the dynamic pressure is of order tex2html_wrap_inline4467 but this is not so as was recently discovered by Kremer & Müller [27]. Indeed, the second step in the Maxwell iteration for tex2html_wrap_inline3395 provides a term that is of order tex2html_wrap_inline4471, 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 (104Popup Equation, 105Popup Equation) can also be calculated for degenerate gases with the thermal equation of state tex2html_wrap_inline4475 for such gases. That equation was also derived by Jüttner [23]. The results for 14 fields may be found in Müller & Ruggeri [39Jump To The Next Citation Point In The Article, 40Jump To The Next Citation Point In The Article].



5.6 Characteristic Speeds in a 5 Relativistic Thermodynamics of Gases. 5.4 The Laws of Navier-Stokes

image 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