A geometric conjecture about phase transitions

A widely accepted view classifies phase transitions based on the type of singularity observed in the relevant thermodynamic potential or its derivatives. While these singularities have been associated with spontaneous symmetry breaking phenomena, the Topological Hypothesis claims that changes to the configuration space topology are the underlying reason for these singularities. This work instead investigates whether changes to the configuration space geometry are more directly related to the onset of a phase transition. We specifically conjecture that first-order phase transitions occur when there is a discontinuity in the configuration space diameter as measured by the mixing time. This conjecture is tested on model systems consisting of hard disks in two dimensions and hard spheres in three dimensions. The diameters of the configuration spaces for increasing numbers of disks/spheres are measured using both the diffusion distance and the mixing time. It is observed that as the number of disks/spheres increases, discontinuities in the diameter of the configuration spaces occur at packing fractions that approach the values reported in the literature for the solid-liquid phase transitions.