It is shown in this section that the concavity of the entropy density with respect to the fields implies global invertibility of the map , where is the n-vector of Lagrange multipliers. Also the system of field equations – written in terms of – 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 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 . 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.
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