Geometrical Frustration in Two Dimensions: Idealizations and Realizations of a Hard Disc Fluid in Negative Curvature

Abstract

We examine a simple hard disc fluid with no long range interactions on the two dimensional space of constant negative Gaussian curvature, the hyperbolic plane. This geometry provides a natural mechanism by which global crystalline order is frustrated, allowing us to construct a tractable model of disordered monodisperse hard discs. We extend free area theory and the virial expansion to this regime, deriving the equation of state for the system, and compare its predictions with simulation near an isostatic packing in the curved space. Additionally, we investigate packing and dynamics on triply periodic, negatively curved surfaces with an eye toward real biological and polymeric systems.

arXiv comment14 pages, 8 figures

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• Crystalline Order on Catenoidal Capillary Bridges
• Thermodynamics and structure of simple liquids in the hyperbolic plane
• Simple equation of state for hard disks on the hyperbolic plane
• Hyperuniformity scaling of maximally random jammed packings of two-dimensional binary disks
• Saddle-splay modulus of a particle-laden fluid interface
• Constraint percolation on hyperbolic lattices
• Variations on the Three-Sphere: Laves' Labyrinth Lopped