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

- finite characteristic speeds and
- well-posedness of initial value problems.

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.

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

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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