### 3.1 Concavity of the entropy density

We recall the argument of Section 2.2 concerning concavity and choose the fields to mean the fields
of densities . Thus equation (8), for , leads to
Therefore the concavity of the entropy density in the variables – the negative-definiteness of
– implies global invertibility between the field vector and the Lagrange multipliers
.
The transformation helps us to recognize the structure of the field equations and to find
generic restrictions on the constitutive functions.

Indeed, obviously, with as field vector instead of , or , we may rephrase (8) in the form

where
Thus we have
and
so that the constitutive quantities and result from – defined by equation (19) – through
differentiation. Therefore the vector is called the thermodynamic vector potential.
It follows from equation (20) that

which implies restrictions on the constitutive functions .