Elements of the constitutive theory

2.2 Elements of the constitutive theory

Since, however, the constitutive functions $^F A$ and π are generally not explicitly known, the major task of thermodynamics is the determination of these functions, or at least the restriction of their generality. In simple cases it is possible to reduce the constitutive functions to a few coefficients which may be turned over to the experimentalist for measurement. The formulation and exploitation of such restrictions is the subject of the constitutive theory.

The tools of the constitutive theory are certain universal physical principles which have come to be accepted by the extrapolation of common experience. Above all there are three such principles:

The Entropy Inequality.

The entropy density $h0$ and the entropy flux $ha$ combine to form a four-vector $hA = (ch0,ha)$ , whose divergence $hA,A$ is equal to the entropy production $Σ$ . The four-vector $hA$ and $Σ$ are both constitutive quantities and $Σ$ is assumed non-negative for all thermodynamic processes. Thus we may write $A ˆ A h = h (u)$ , $ˆ Σ = Σ (u)$ and

This inequality is clearly an extrapolation of the entropy inequalities known in thermostatics and thermodynamics of irreversible processes; it was first stated in this generality by Müller [36 , 38 ].

The Principle of Relativity.

The principle of relativity requires that the field equations and the entropy inequality have the same form in all

Galilei frames for the non-relativistic case, or in all
Lorentz frames for the relativistic case.

The formal statement and exploitation of this principle have to await a specific choice for the fields $u$ and the four-fluxes $FA$ .

The Requirement of Concavity of the Entropy Density.

It is possible, and indeed common, to make a specific choice for the fields $u$ and the concavity postulate is contingent upon that choice.

In the non-relativistic case we choose the fields $u$ as the densities $F 0$ . The requirement of concavity demands that the entropy density $h0$ be a concave function of the variables $F 0$ :
In the relativistic case we choose the fields $u$ as the densities $F = F Aζ ζ A$ in a generic Lorentz frame that moves with the four-velocity $cζA$ with respect to the observer. We have $A ζ ζA = 1$ and $0 ζ > 0$ . We cannot be certain that in all these frames the entropy density $hζ = hA ζA$ is concave as a function of $F ζ$ . Therefore we assume that there is at least one $ζA$ – a privileged one, denoted by $¯ζA$ – such that $hζ¯= hA ¯ζA$ is concave with respect to $A F ¯ζ = F ζA$ , viz.

The privileged co-vector $¯ ζA$ remains to be chosen, see Section 4.1.

In both cases the concavity postulate makes it possible that the entropy be maximal for a particular set of fields – the set corresponding to equilibrium – and that is its attraction for physicists. For mathematicians the attraction of the concavity postulate lies in the observation that concavity implies symmetric hyperbolicity of the field equations, see Sections 3.2 and 4.2 below.