Concavity of the entropy density

3.1 Concavity of the entropy density

We recall the argument of Section 2.2 concerning concavity and choose the fields $u$ to mean the fields of densities $0 F$ . Thus equation (8

), for $A = 0$ , leads to

Therefore the concavity of the entropy density $h0$ in the variables $F 0$ – the negative-definiteness of $∂2h0 ----0---0 ∂F ∂F$ – implies global invertibility between the field vector $F 0$ and the Lagrange multipliers $Λ$ .

The transformation $0 F ⇐ ⇒ Λ$ helps us to recognize the structure of the field equations and to find generic restrictions on the constitutive functions.

Indeed, obviously, with $Λ$ as field vector instead of $u$ , or $F 0$ , we may rephrase (8 ) in the form

where

Thus we have

and

so that the constitutive quantities $F A$ and $hA$ result from $h′A$ – defined by equation (19

) – through differentiation. Therefore the vector $′A h$ is called the thermodynamic vector potential.

It follows from equation (20 ) that

A ∂F--- is symmetric, ∂Λ

which implies $4n (n − 1)$ restrictions on the constitutive functions $^F A(Λ )$ .