A Theorem on the origin of Phase Transitions

Abstract

For physical systems described by smooth, finite-range and confining microscopic interaction potentials V with continuously varying coordinates, we announce and outline the proof of a theorem that establishes that unless the equipotential hypersurfaces of configuration space Σ_v ={(q_1,...,q_N)\in R^N | V(q_1,...,q_N) = v}, v \in R, change topology at some v_c in a given interval [v_0, v_1] of values v of V, the Helmoltz free energy must be at least twice differentiable in the corresponding interval of inverse temperature (β(v_0), β(v_1)) also in the N -> \infty$ limit. Thus the occurrence of a phase transition at some β_c =β(v_c) is necessarily the consequence of the loss of diffeomorphicity among the {Σ_v}_{v < v_c}$ and the {Σ_v}_{v > v_c}, which is the consequence of the existence of critical points of V on Σ_{v=v_c}, that is points where \nabla V=0.