Requirements for Stability

We investigate whether a homogeneous system is stable with respect to breakup into a composite system of two or more homogeneous subsystems. Criteria to avoid breakup lead to requirements for the dependence of the entropy and thermodynamic potentials on their natural variables. For stability, the entropy must be a concave function of its natural variables (all extensive) and the internal energy must be a convex function of its natural variables (all extensive). The thermodynamic potentials (Helmholtz, enthalpy, Gibbs, Kramers) must be convex functions of their extensive variables and concave functions of their intensive variables. Properties of Legendre transformations are used to derive the stability requirements for intensive variables. Local stability criteria depend on the signs of second order partial derivatives. When these stability criteria are violated, there can be locally unstable regions and metastable regions that are locally stable but globally unstable. Then transformations can occur. Principles of Le Chatlier and Le Chatlier-Braun elucidate the approach to equilibrium.