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

TL;DR

This paper explores how negative curvature affects the behavior of hard disc fluids in two dimensions, revealing insights into disorder, packing, and dynamics relevant to biological and polymeric systems.

Contribution

It introduces a tractable model of disordered hard discs on negatively curved surfaces and extends theoretical methods to analyze their thermodynamics and packing.

Findings

Equation of state derived for hard discs on hyperbolic plane

Comparison shows good agreement between theory and simulation near isostatic packing

Insights into packing and dynamics on negatively curved surfaces

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.