A heuristic theory of phase transitions

Abstract

LetZ be a suitable Banach space of interactions for a lattice spin system. Ifn+1 thermodynamic phases coexist for Φ₀ ∈Z, it is shown that a manifold of codimensionn of coexistence of (at least)n+1 phases passes through Φ₀. There are alson+1 manifolds of codimensionn−1 of coexistence of (at least)n phases; these have a common boundary along the manifold of coexistence ofn+1 phases. And so on for coexistence of fewer phases. This theorem is proved under a technical condition (R) which says that the pressure is a differentiable function of the interaction at Φ₀ when restricted to some codimensionn affine subspace ofZ. The condition (R) has not been checked in any specific instance, and it is possible that our theorem is useless or vacuous. We believe however that the method of proof is physically correct and constitutes at least a heuristic proof of the Gibbs phase rule.