3 Finite Speeds in Non-Relativistic Extended Thermodynamics

It is shown in this section that the concavity of the entropy density h0 with respect to the fields F 0 implies global invertibility of the map 0 F ⇐ ⇒ Λ, where Λ is the n-vector of Lagrange multipliers. Also the system of field equations – written in terms of Λ – is recognized as a symmetric hyperbolic system which guarantees

Thus we conclude that no paradox of infinite speeds can arise in extended thermodynamics, – at least not for finitely many variables.

A commonly treated special case occurs when the fields u are moments of the phase density of a gas. In this case the pulse speed depends on the degree of extension, i.e. on the number n of fields u. For a gas in equilibrium the pulse speeds can be calculated for any n. Also it can be estimated that the pulse speed tends to infinity as n grows to infinity.

 3.1 Concavity of the entropy density
 3.2 Symmetric hyperbolicity
 3.3 Moments as variables
 3.4 Specific form of the phase density
 3.5 Pulse speeds in a non-degenerate gas in equilibrium
 3.6 A lower bound for the pulse speed of a non-degenerate gas

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