A Generalized Thermodynamic Criterion of Equilibrium

Thermodynamic functions, such as entropy, underlie macroscopic and microscopic thermodynamics, and supply essential tools to many related disciplines. Determining a steady-state process is one of the basic roles of the thermodynamic criterion of equilibrium. However, the classical thermodynamic criterion of equilibrium requires second-order differentiable functions, that is, a strong constraint in mathematics. To relax this constraint, convex analysis is applied to reformulate a generalized thermodynamic criterion of equilibrium only with first-order differentiable requirement. Then, it is proved that this generalized stability condition is a necessary but insufficient condition for the classical one. In addition to isolated systems, the corresponding criteria under various constraints are given. This theorem works in some thermodynamic systems which are only first-order differentiable, such as second-order phase-transitions. To indicate that this theorem determines the direction of spontaneous change and the stability of equilibrium, different thermodynamic problems are demonstrated with the generalized criterion, thereby revealing the usefulness of this theorem.