On non-Poissonian Voronoi tessellations

TL;DR

This paper investigates Voronoi tessellations generated by Sobol quasi-random sequences, proposing a volume distribution, analyzing chord lengths, and introducing a new generalized tessellation model.

Contribution

It introduces Sobol-based Voronoi tessellations, derives their volume distribution, analyzes chord length distributions, and proposes a new generalized tessellation model.

Findings

Proposed a probability density function for Sobol Voronoi tessellations.

Validated analytical formulas with numerical simulations.

Presented an application to gas structure analysis.

Abstract

The Voronoi tessellation is the partition of space for a given seeds pattern and the result of the partition depends completely on the type of given pattern "random", Poisson-Voronoi tessellations (PVT), or "non-random", Non Poisson-Voronoi tessellations. In this note we shall consider properties of Voronoi tessellations with centers generated by Sobol quasi random sequences which produce a more ordered disposition of the centers with respect to the PVT case. A probability density function for volumes of these Sobol Voronoi tessellations (SVT) will be proposed and compared with results of numerical simulations. An application will be presented concerning the local structure of gas ($CO_2$) in the liquid-gas coexistence phase. Furthermore a probability distribution will be computed for the length of chords resulting from the intersections of random lines with a three-dimensional SVT. The agreement of the analytical formula with the results from a computer simulation will be also investigated. Finally a new type of Voronoi tessellation based on adjustable positions of seeds has been introduced which generalizes both PVT and SVT cases.