Quotient Maps and Configuration Spaces of Hard Disks

TL;DR

This paper investigates how quotient maps affect the topology and geometry of configuration spaces of hard disks on tori, with implications for understanding phase transitions and glassy systems.

Contribution

It provides explicit constructions of configuration space triangulations and analyzes how symmetry quotients influence topological features and critical points.

Findings

Topology of configuration spaces depends on symmetry quotients.

Critical points relate to changes in topology and entropy.

Explicit triangulations of configuration spaces are constructed.

Abstract

Hard disks systems are often considered as prototypes for simple fluids. In a statistical mechanics context, the hard disk configuration space is generally quotiented by the action of various symmetry groups. The changes in the topological and geometric properties of the configuration spaces effected by such quotient maps are studied for small numbers of disks on a square and hexagonal torus. A metric is defined on the configuration space and the various quotient spaces that respects the desired symmetries. This is used to construct explicit triangulations of the configuration spaces as $\alpha$-complexes. Critical points in a configuration space are associated with changes in the topology as a function of disk radius, are conjectured to be related to the configurational entropy of glassy systems, and could reveal the origins of phase transitions in other systems. The number and topological and geometric properties of the critical points are found to depend on the symmetries by which the configuration space is quotiented.