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 (8View Equation), for A = 0, leads to
∂h0 ∂ Λ ∂2h0 Λ = ----0, hence ---0-= ---0---0. (17 ) ∂F ∂F ∂F ∂F
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 (8View Equation) in the form

dh′A = F AdΛ, (18 )
h′A ≡ Λ ⋅ F A − hA. (19 )
Thus we have
∂h ′A F A = -----, (20 ) ∂Λ
′A hA = Λ ∂h---− h′A, (21 ) ∂Λ
so that the constitutive quantities F A and hA result from h′A – defined by equation (19View Equation) – through differentiation. Therefore the vector ′A h is called the thermodynamic vector potential.

It follows from equation (20View Equation) that

A ∂F--- is symmetric, ∂Λ

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

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