Fundamental functions in equilibrium thermodynamics

In the standard presentations of the principles of Gibbsian equilibrium thermodynamics one can find several gaps in the logic. For a subject that is as widely used as equilibrium thermodynamics, it is of interest to clear up such questions of mathematical rigor. In this paper it is shown that using convex analysis one can give a mathematically rigorous treatment of several basic aspects of equilibrium thermodynamics. On the basis of a fundamental convexity property implied by the second law, the following topics are discussed: thermodynamic stability, transformed fundamental functions (such as the Gibbs free energy), and the existence and uniqueness of possible final equilibrium states of closed composite thermodynamic systems. It is shown that a standard mathematical characterization of thermodynamic stability (involving a positive definiteness property) is sufficient but in fact not necessary for the physically superior convexity characterization of thermodynamic stability. Furthermore, it is found that functions such as the Gibbs free energy can be rigorously and globally defined using convex conjugation instead of Legendre transformation. Another result desribed in this paper is that equilibrium thermodynamics cannot always uniquely predict possible final equilibrium states of closed composite thermodynamic systems