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

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Voronoi Entropy (VE) quantifies order in point patterns, ranging from 0 for ordered sets to 1.69 for random sets. VE captures long-range order, unlike Shannon Entropy, and correlates with system correlations.

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Area of Science:

  • Statistical physics
  • Computational geometry
  • Materials science

Background:

  • Voronoi Entropy (VE) is a measure of disorder in 2D point processes.
  • Hyperuniformity describes point sets with suppressed long-range density fluctuations.
  • Understanding the relationship between VE and hyperuniformity is crucial for characterizing complex systems.

Purpose of the Study:

  • To investigate the asymptotic maximum value and convergence of VE for random and hyperuniform point processes.
  • To determine the relationship between VE and parameters like the number of polygons and region size.
  • To compare VE's ability to capture long-range order with Shannon Entropy.

Main Methods:

  • Calculation of VE for 2D random point processes and hyperuniform point sets.
  • Analysis of VE's range (0 to 1.69) based on point set order.
  • Identification of critical radii (Limit-1 and Limit-2) for Voronoi diagram construction and VE saturation.

Main Results:

  • VE ranges from 0 (ordered) to 1.69 (random) for n > 100 polygons.
  • Limit-1 (R=2.5) is the minimum radius for Voronoi diagram construction; Limit-2 (R=5.5) marks VE saturation.
  • VE exceeds 1.69 in some seed point patterns and correlates with long-range correlations in hyperuniform systems.

Conclusions:

  • VE is a sensitive indicator of order and long-range correlations in 2D point patterns.
  • VE captures aspects of long-range order missed by Shannon Entropy due to geometric constraints.
  • The study provides insights into the behavior of VE in systems ranging from ordered to random, including hyperuniform materials.