GENERIC ASPECTS OF CONVEXIFICATION WITH APPLICATIONS TO THERMODYNAMIC-EQUILIBRIUM

This work presents a mathematical analysis of the process of convexification of a smooth function based on singularity theory. The theory developed is applied to the problem of thermodynamic phase equilibrium. The central notion introduced here is that of phase simplex, which we use to discuss phase equilibrium and phase transition in an abstract framework. One of the by-products of the results of this paper is a rigorous proof of Gibbs' Phase Rule for multicomponent systems, in which a well-accepted mathematical notion of genericity is used to account for ostensible exceptions to the rule. Also, many other features known from theoretical or experimental thermodynamics can be rediscovered through purely mathematical arguments from the notions introduced here. Such features include, among other things, the existence of saturation pressures, the existence of multiple or critical points, and the existence of spontaneous or continuous changes in the composition of the phases at phase transitions.