Extreme Values and Convergence of the Voronoi Entropy for 2D Random Point Processes and for Long-Range Order

We investigate the asymptotic maximum value and convergence of the Voronoi Entropy (VE) for a 2D random point process (S = 1.690 ± 0.001) and point sets with long-range order characterized by hyperuniformity. We find that for the number of polygons of about n > 100, the VE range is between S = 0 (ordered set of seed points) and S = 1.69 (random set of seed points). For circular regions with the dimensionless radius R normalized by the average distance between points, we identify two limits: Limit-1 (R = 2.5, 16 ± 6 points) is the minimum radius, for which it is possible to construct a Voronoi diagram, and Limit-2 (R = 5.5, 96 ± 6 points) at which the VE reaches the saturation level. We also discuss examples of seed point patterns for which the values of VE exceed the asymptotic value of S > 1.69. While the VE accounts only for 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.

• Publication year

  2026

• Venue

  Entropy

• Publication date

  2026-01-01

• Fields of study

  Mathematics, Physics, Computer Science, Medicine

• Identifiers
  DOI 10.3390/e28010095PMID 41594002PMCID 12840345
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar, PubMed

• No claims are published for this paper.

• No concepts are published for this paper.

• Crystallization Control of Anionic Thiacalixarenes on Silicon Surface Coated with Cationic Poly(ethyleneimine).

  2024cited by this paper
• Converting Tessellations into Graphs: From Voronoi Tessellations to Complete Graphs

  2024influential reference
• Voronoi Diagrams Generated by the Archimedes Spiral: Fibonacci Numbers, Chirality and Aesthetic Appeal

  2023cited by this paper
• Voronoi Entropy as a Ligand Molecular Descriptor of Protein–Ligand Interactions

  2023cited by this paper
• Topological Data Analysis of Nanoscale Roughness of Layer-by-Layer Polyelectrolyte Samples Using Machine Learning

  2023cited by this paper
• Aperiodic crystals, Riemann zeta function, and primes

  2022cited by this paper
• Continuous Symmetry Measure vs Voronoi Entropy of Droplet Clusters

  2021cited by this paper
• Aperiodic crystals in biology

  2021cited by this paper
• Analysis of the Number of Sides of Voronoi Polygons in PassPoint

  2021influential reference
• Clustering and self-organization in small-scale natural and artificial systems

  2020cited by this paper
• Random sequential adsorption of particles with tetrahedral symmetry.

  2019cited by this paper
• Droplet clusters: nature-inspired biological reactors and aerosols

  2019cited by this paper
• Is the Voronoi Entropy a True Entropy? Comments on “Entropy, Shannon’s Measure of Information and Boltzmann’s H-Theorem”, Entropy 2017, 19, 48

  2019cited by this paper
• Hyperuniform states of matter

  2018cited by this paper
• Characterization of Self-Assembled 2D Patterns with Voronoi Entropy

  2018influential reference
• Entropy, Shannon's Measure of Information and Boltzmann's H-Theorem

  2017influential reference
• Ground states of stealthy hyperuniform potentials. II. Stacked-slider phases.

  2015cited by this paper
• Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations.

  2015cited by this paper
• Entropy‐Driven Phase Transitions in Colloids: From spheres to anisotropic particles

  2014influential reference
• Fifty years of aperiodic crystals.

  2012cited by this paper
• Non-crystalline colloidal clusters in two dimensions: size distributions and shapes

  2012cited by this paper
• A Mathematical Theory of Communication

  2006cited by this paper
• On the true and apparent densities of nanoparticles

  2005cited by this paper
• Constraints on collective density variables: two dimensions.

  2004influential reference
• Quasicrystals: A Matter of Definition

  2003cited by this paper
• An explicit expression for the distribution of the number of sides of the typical Poisson-Voronoi cell

  2003cited by this paper
• Statistical Distributions of Poisson Voronoi Cells in Two and Three Dimensions

  2003influential reference
• The geometrical properties of irregular two-dimensional Voronoi tessellations

  2001cited by this paper
• The proportion of triangles in a Poisson-Voronoi tessellation of the plane

  2000cited by this paper
• Evidence for convective effects in breath figure formation on volatile fluid surfaces.

  1996influential reference
• Symmetry as a Continuous Feature

  1995influential reference
• Properties of a two-dimensional Poisson-Voronoi tesselation: A Monte-Carlo study

  1993influential reference
• Metallic Phase with Long-Range Orientational Order and No Translational Symmetry

  1984cited by this paper
• Monte carlo estimates of the distributions of the random polygons of the voronoi tessellation with respect to a poisson process

  1980influential reference
• The Monte-Carlo generation of random polygons

  1978influential reference

• No citing papers are available for this paper.