Specific results for a non-degenerate relativistic gas

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

fE = ----[------y-----]----- or fE = ---⌊-----------y∘----------⌋----. (100 ) exp m-α + UA-pA ∓ 1 m mc2 p2 k kT exp⌈ --α + ---- 1 + --22-⌉ ∓ 1 k kT m c

The latter equation is valid in the rest frame of the gas. $pA$ is the atomic four-momentum and we have $A 2 2 pAp = m c$ . 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 $p = p(α, T)$ reads

where $1∕y$ is the smallest phase space element. From (102

) we obtain with $G = K3- K2$ and $2 γ = mc-- kT$

and hence

( ) C0 = nm 1 + 6 G 1 ( γ) ( ) ( ) 2− 5- + 19− 30- G− 2− 45- G2− 9G3 Cπ1 = − 62----γ2---(γ--γ3)-------γ2-----γ-- c 3γ− 2− 2γ02 G− 13γ G2+2G3 (104 ) 11+6γG− G2 C3 = − γ1+5γG−-G2 ( 6 1) C5 = γ + G- .

The transport coefficients read

( ) 1nmc2-1-−γ3+-2− 2γ02-G+-1γ3G2+2G3 λ = 3 γ B1π 1−γ12+5γG −G2 1-- (105 ) μ = − nkT B3G c2- 2 κ = − nkT ˆB4 (γ + 5G − γG ).

It is instructive to calculate the leading terms of the transport coefficients in the non-relativistic case $2 mc ≫ kT$ . 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 $λ$ , $μ$ and $κ$ 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 $p = p(α, T)$ and the $B$ ’s may be measured.

It might seem from (106 ) and (97 ) that the dynamic pressure is of order $( ) 1 O --2 γ$ but this is not so as was recently discovered by Kremer & Müller [27 ]. Indeed, the second step in the Maxwell iteration for $π$ provides a term that is of order $( ) 1- O γ$ , 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 $p(α, T)$ for such gases. That equation was also derived by Jüttner [23 ]. The results for 14 fields may be found in Müller & Ruggeri [39 , 40 ].