A geometric conjecture about phase transitions

TL;DR

This paper proposes that geometric changes in the configuration space of particle systems drive phase transitions, supported by numerical evidence linking mixing time discontinuities to solid-fluid transitions.

Contribution

It introduces a geometric conjecture connecting phase transitions to changes in configuration space geometry, tested through explicit geometries and numerical analysis.

Findings

Discontinuity in mixing time coincides with phase transition

Explicit configuration space geometries constructed for hard particles

Numerical evidence supports the geometric conjecture

Abstract

As phenomena that necessarily emerge from the collective behavior of interacting particles, phase transitions continue to be difficult to predict using statistical thermodynamics. A recent proposal called the topological hypothesis suggests that the existence of a phase transition could perhaps be inferred from changes to the topology of the accessible part of the configuration space. This paper instead suggests that such a topological change is often associated with a dramatic change in the configuration space geometry, and that the geometric change is the actual driver of the phase transition. More precisely, a geometric change that brings about a discontinuity in the mixing time required for an initial probability distribution on the configuration space to reach steady-state is conjectured to be related to the onset of a phase transition in the thermodynamic limit. This conjecture is tested by evaluating the diffusion diameter and $\epsilon$-mixing time of the configuration spaces of hard disk and hard sphere systems of increasing size. Explicit geometries are constructed for the configuration spaces of these systems, and numerical evidence suggests that a discontinuity in the $\epsilon$-mixing time coincides with the solid-fluid phase transition in the thermodynamic limit.