Configuration spaces of hard spheres

TL;DR

This paper investigates the topological and geometric properties of configuration spaces of hard spheres, revealing their relation to phase transitions and providing a comprehensive analysis through Morse theory and explicit triangulations.

Contribution

It introduces a Morse-theoretic approach to sample critical configurations and constructs explicit triangulations for two-sphere systems, linking topological changes to phase transition phenomena.

Findings

Critical configurations connect distant regions in configuration space.

Number of critical configurations grows exponentially with sphere count.

Configuration space diameter discontinuity correlates with phase transition onset.

Abstract

Hard sphere systems are often used to model simple fluids. The configuration spaces of hard spheres in a three-dimensional torus modulo various symmetry groups are comparatively simple, and could provide valuable information about the nature of phase transitions. Specifically, the topological changes in the configuration space as a function of packing fraction have been conjectured to be related to the onset of first-order phase transitions. The critical configurations for one to twelve spheres are sampled using a Morse-theoretic approach, and are available in an online, interactive database. Explicit triangulations are constructed for the configuration spaces of the two sphere system, and their topological and geometric properties are studied. The critical configurations are found to be associated with geometric changes to the configuration space that connect previously distant regions and reduce the configuration space diameter as measured by the commute time and diffusion distances. The number of such critical configurations around the packing fraction of the solid-liquid phase transition increases exponentially with the number of spheres, suggesting that the onset of the first-order phase transition in the thermodynamic limit is associated with a discontinuity in the configuration space diameter.