Quotient maps and configuration spaces of hard disks

• Published in: Granular matter Vol. 24; no. 3
• Main Authors: Ericok, Ozan B., Mason, Jeremy K.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2022; Springer Nature B.V
• Subjects: Complex Fluids and Microfluidics; Configurations; Critical point; Disk drives; Disks; Engineering Fluid Dynamics; Engineering Thermodynamics; Foundations; Geoengineering; Hard disks; Heat and Mass Transfer; Hydraulics; Industrial Chemistry/Chemical Engineering; Materials Science; Original Paper; Phase transitions; Physics; Physics and Astronomy; Quotients; Soft and Granular Matter; Statistical mechanics; Topology; Toruses; Geometry; Hard disks; Configuration spaces; Topology; Phase transitions
• ISSN: 1434-5021; 1434-7636; 1434-7636
• DOI: 10.1007/s10035-022-01235-5

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 α -complexes. Critical points of the hard disk potential on a configuration space are associated with changes in the topology of the accessible part of the configuration space 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 of critical points and their topological and geometric properties are found to depend on the symmetries by which the configuration space is quotiented. Graphic abstract