A Stability Criterion for Many Parameter Equilibrium Families

Theorems are established which let one detect instabilities without recourse to the usual perturbation analysis. The method applies to any system whose stable equilibria maximize a functional S at fixed values of one or more parameters E^α. It generalizes the "turning point method" by inferring instability from the behavior in equilibrium of the E^α and of their conjugate parameters ∂S/∂SE^α. The "cusp catastrophe" and the black hole equilibrium family illustrate the approach. In connection with the latter, an Appendix proves that the Gibbs free energy is an analytic function of its natural arguments, as would be expected if all the equilibria belonged to a single thermodynamic phase.