A Theorem on the origin of Phase Transitions

TL;DR

This paper proves a theorem linking phase transitions in physical systems to topological changes in the configuration space, specifically the loss of diffeomorphicity caused by critical points of the potential.

Contribution

It establishes a rigorous connection between phase transitions and topological changes in the energy landscape of finite-range, confining potentials.

Findings

Phase transitions correspond to topology changes in configuration space.

Critical points of the potential induce topological changes leading to phase transitions.

The Helmholtz free energy's differentiability is linked to the topology of equipotential hypersurfaces.

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 \Sigma_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 (\beta(v_0), \beta(v_1)) also in the N -> \infty$ limit. Thus the occurrence of a phase transition at some \beta_c =\beta(v_c) is necessarily the consequence of the loss of diffeomorphicity among the {\Sigma_v}_{v < v_c}$ and the {\Sigma_v}_{v > v_c}, which is the consequence of the existence of critical points of V on \Sigma_{v=v_c}, that is points where \nabla V=0.