3.5 Pulse speeds in a non-degenerate gas in equilibrium

We recall the discussion of characteristic speeds in Section 2.4 which we apply to the system (23View Equation) of field equations. The characteristic equation of this system reads
( ) -∂2h′A- det ϕ,A∂Λ ∂ Λ = 0 (45 )
or, by (11View Equation):
( 2 ′a 2 ′0 ) det -∂-h---na − V--∂-h--- = 0. (46 ) ∂ Λ∂ Λ c ∂ Λ∂ Λ
This equation determines the characteristic speeds V, whose maximal value Vmax is the pulse speed. In the case of moments and for a non-degenerate gas at rest and in equilibrium this equation reads, by (42View Equation),
(∫ a ) det (p na − V m )pαp βfEdp = 0. (47 )
fE is the Maxwellian phase density, so that all integrals in (47View Equation) are Gaussian integrals, easy to calculate. Weiss [49] has calculated the speeds V for different degrees n of extended thermodynamics. Recall that α, β range over the values 1 through n. He has made a list of Vmax which is represented here in Table 1. Vmax is normalized in Table 1 by ∘ 5kT- co = 3m, the ordinary speed of sound, sometimes called the adiabatic sound speed.

Inspection of Table 1 shows that the pulse speed increases monotonically with the number of moments and there is clearly a suspicion that it may tend to infinity as n goes to infinity. This suspicion will presently be confirmed.


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