Topology and Phase Transitions I. Preliminary Results

TL;DR

This paper proves a theorem linking the topology of certain configuration space submanifolds to the occurrence of phase transitions, establishing a necessary topological condition for first and second order phase transitions.

Contribution

It introduces a theorem showing that topology changes in configuration space are necessary for phase transitions, advancing the understanding of topological conditions in statistical mechanics.

Findings

Topology of submanifolds must change at phase transition points.

Equipotential hypersurfaces are diffeomorphic in the absence of phase transitions.

Helmoltz free energy converges uniformly in the thermodynamic limit under certain conditions.

Abstract

In this first paper, we demonstrate a theorem that establishes a first step toward proving a necessary topological condition for the occurrence of first or second order phase transitions: we prove that the topology of certain submanifolds of configuration space must necessarily change at the phase transition point. The theorem applies to smooth, finite-range and confining potentials V bounded below, describing systems confined in finite regions of space with continuously varying coordinates. The relevant configuration space submanifolds are both the level sets {\Sigma_v := V_N^{-1} (v)}_{v \in R} of the potential function V_N and the configuration space submanifolds enclosed by the \Sigma_{v} defined by {M_v := V_N^{-1} ((-\infty,v])}_{v \in R}, which are labeled by the potential energy value v, and where N is the number of degrees of freedom. The proof of the theorem proceeds by showing that, under the assumption of diffeomorphicity of the equipotential hypersurfaces {\Sigma_v}_{v \in R}, as well as of the ${M_v}_{v \in R}, in an arbitrary interval of values for \vb=v/N, the Helmoltz free energy is uniformly convergent in N to its thermodynamic limit, at least within the class of twice differentiable functions, in the corresponding interval of temperature. This preliminary theorem is essential to prove another theorem - in paper II - which makes a stronger statement about the relevance of topology for phase transitions.