Finite Speeds in Non-Relativistic Extended Thermodynamics

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

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 $u$ 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