Computational topology for configuration spaces of hard disks

Abstract

We explore the topology of configuration spaces of hard disks experimentally, and show that several changes in the topology can already be observed with a small number of particles. The results illustrate a theorem of Baryshnikov, Bubenik, and Kahle that critical points correspond to configurations of disks with balanced mechanical stresses, and suggest conjectures about the asymptotic topology as the number of disks tends to infinity.

version 3: 16 pages, 11 figures; made minor changes requested by journal --- changed some color figures to black and white, and reordered bibliography to appear in order of citation rather than alphabetical order

References in corpus (5)

• Topological Methods for Exploring Low-density States in Biomolecular Folding Pathways
• Robust Algorithm to Generate a Diverse Class of Dense Disordered and Ordered Sphere Packings via Linear Programming
• Min-type Morse theory for configuration spaces of hard spheres
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Gibbs/Metropolis algorithms on a convex polytope

Cited by in corpus (22)

• Principal Component Analysis of Persistent Homology Rank Functions with case studies of Spatial Point Patterns, Sphere Packing and Colloids
• Calculations of the Structure of Basin Volumes for Mechanically Stable Packings
• Exploring the energy landscape of XY models
• Characterizing Granular Networks Using Topological Metrics
• Topological Theory of Phase Transitions
• Shapes within shapes: how particles arrange inside a cavity
• Geometrical aspects in the analysis of microcanonical phase-transitions
• Transition state theory and the dynamics of hard disks
• Configuration spaces of disks in a strip, twisted algebras, persistence, and other stories
• Homology of configuration spaces of hard squares in a rectangle
• Configuration spaces of hard spheres
• Visualizing Free Energy Landscapes for Four Hard Disks
• What is liquid? Lyapunov instability reveals symmetry-breaking irreversibilities hidden within Hamilton's many-body equations of motion
• The configurational entropy of colloidal particles in a confined space
• Quotient Maps and Configuration Spaces of Hard Disks
• Calculating the symmetry number of flexible sphere clusters
• How the overlap of excluded volume determines the configurational energy landscape and "thermodynamics" in the "one to five hard disks in a box" system
• Generalized representation stability for disks in a strip and no-k-equal spaces
• Leaky Cell Model of Hard Spheres
• A geometric conjecture about phase transitions
• Atlasing of Assembly Landscapes using Distance Geometry and Graph Rigidity
• Toward a refining of the topological theory of phase transitions