### 2.3 Exploitation of the entropy inequality, Lagrange multipliers

The key to the exploitation of the entropy inequality lies in the fact that the inequality should hold for thermodynamic processes, i.e. solutions of the field equations rather than for all fields. By a theorem proved by Liu [30] this constraint may be removed by the use of Lagrange multipliers – themselves constitutive quantities, so that holds. Indeed, the new inequality
is equivalent to (3).

Liu’s proof proceeds from the observation that the field equations and the entropy equation are linear functions of the derivatives . By the Cauchy–Kowalewski theorem these derivatives are local representatives of an analytical thermodynamic process and therefore the entropy principle requires that the field equations and the entropy equation must hold for all . It is then a simple problem of linear algebra to prove that

Liu’s proof is not restricted to quasilinear systems of first order equations but here we need his result only in that particularly simple case.

We may use the chain rule on and in (6) and obtain

The left hand side is an explicit linear function of the derivatives and, since the inequality must hold for all fields , it must hold in particular for arbitrary values of the derivatives . The entropy inequality could thus easily be violated by some choice of unless we have
and there remains the residual inequality

The differential forms (8) represent a generalization of the Gibbs equation of equilibrium thermodynamics; the classical Gibbs equation for the entropy density is here generalized into four equations for the entropy four-flux. Relation (9) is the residual entropy inequality which represents the irreversible entropy production. Note that the entropy production is entirely due to the production terms in the balance equations.