5.5 Specific results for a non-degenerate relativistic gas

For a relativistic gas Jüttner [2223Jump To The Next Citation Point] 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.
( ) ( ) mc2 ∫∞ mc2 Kn ---- = cos h(nρ) exp − ---- coshρ d ρ (101 ) kT 0 kT
are the relevant special functions. The thermal equation of state p = p(α, T) reads
( ) ( mc2) m- 3 3K2---kT-- p = nkT with n = exp − k α ⋅ 4πym c mc2 , (102 ) kT
where 1∕y is the smallest phase space element. From (102View Equation) we obtain with G = K3- K2 and 2 γ = mc-- kT
( ) e = nmc2 G − -1 , Γ-1= nmc2 2-G (103 ) γ T γ
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
λ = − -5--nkT -1- (106 ) 6B1π γ2
1 μ = − ---nkT (107 ) B3
-5--nk2T-- κ = − 2Bˆ m . (108 ) 4
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 (105View 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 p = p(α, T) and the B’s may be measured.

It might seem from (106View Equation) and (97View Equation) 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 (104View Equation, 105View Equation) 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 [39Jump To The Next Citation Point40Jump To The Next Citation Point].


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