Symmetries and Hard Disk Configuration Spaces

Hard disks systems are often considered as prototypes for simple fluids. Critical points (mechanically-balanced configurations) in the hard disk configuration space are involved in estimating the configurational entropy, and could contain valuable information about the origins of phase transitions. Since configuration spaces generally contain multiple configurations related by physical symmetries, they are usually quotiented by the action of these symmetries. A metric respecting the desired physical symmetries is proposed, and quotient maps that transform the configuration space into the quotient space are examined. The original and quotiented configuration spaces are explicitly triangulated, and the topological and geometric properties are studied for small number of disks on a square and hexagonal torus. It is found that the number of the critical points depends in unexpected ways on the application of physical symmetries, with implications for the apparent configurational entropy and mixing time of these systems.