Voronoi Entropy and long-range order of 2D point sets

TL;DR

This paper introduces Voronoi Entropy as a novel measure of order in 2D point sets, capturing both local and long-range order properties, and compares it with long-range correlation metrics.

Contribution

It proposes Voronoi Entropy as a new quantitative tool for analyzing the orderliness and long-range correlations in 2D point distributions, especially hyperuniform systems.

Findings

VE ranges from 0 for perfect order to ~1.69 for randomness.

VE correlates with long-range density exponents up to a saturation point.

VE decreases to zero as long-range order diminishes.

Abstract

The Voronoi Entropy (VE) and the continuous measure of symmetry (CSM) characterize the orderliness of a set of points on a 2D plane. The Voronoi entropy is the Shannon entropy of the Voronoi tessellation of the plane into polygons, quantifying the diversity of polygons. The VE is widely used to study the self-assembly of colloidal, supramolecular, and other systems. The value of VE changes from S=0 for a completely ordered system, built of polygons with an equal number of sides, to S=1.690+/-0.001 for a random set of points. While the VE takes into account only neighboring polygons, covering the 2D plane imposes constraints on the number of polygons and the number of edges in polygons. Consequently, unlike the conventional Shannon Entropy, the VE captures some long-range order properties of the system. We calculate the VE for several hyperuniform sets of points and compare it with the values of exponents of collective density variables characterizing long-range correlations in the system. We show that the VE correlates with the latter up to a certain saturation level, after which the value of the VE falls to S=0, and we explain this phenomenon.