Statistical topology of perturbed two-dimensional lattices

Abstract

The Voronoi cell of any atom in a lattice is identical. If atoms are perturbed from their lattice coordinates, then the topologies of the Voronoi cells of the atoms will change. We consider the distribution of Voronoi cell topologies in two-dimensional perturbed systems. These systems can be thought of as simple models of finite-temperature crystals. We give analytical results for the distribution of Voronoi topologies of points in two-dimensional Bravais lattices under infinitesimal perturbations and present a discussion with numerical results for finite perturbations.

Voronoi chains, blocks, and clusters in perturbed square lattices

“…Calculations similar to those presented in Ref. [10] show that the probability of a Voronoi cell being hexagonal of the first kind is 2[(π − θ)/2π] 2 ≈ 8.80%, whereas that of it being hexagonal of the second kind is [θ/π] 2 ≈ 33.69%. Indeed, many more hexagonal cells of the second kind can be observed in Fig.…”

“…3. As shown previously [10], the events v 00 and v 01 are pairwise independent, as are v 10 and v 11 , and so P (v 00 , v 01 ) = P (v 00 )P (v 01 ) = 1 4 , and P (v 10 , v 11 ) = P (v 10 )P (v 11 ) = 1 4 . We therefore have:…”

“…Statistical topology has been previously used to study glasses [1], polymers [2], polycrystalline metals [3], radial and cellular networks [4,5], ideal gases [6], and supercritical fluids [7,8,9]. This paper builds on earlier work that computed the distribution of Voronoi topologies in perturbed two-dimensional crystals [10]. The authors illustrated how simple probabilistic analysis can be used to determine the distribution of Voronoi cells, as classified by their number of edges, in such systems.…”

Voronoi cell analysis: The shapes of particle systems

“…Small perturbations of the particles, which can be seen as representative of thermal vibrations or measurement errors, result in Voronoi cells that are topologically distinct from those in the unperturbed case. 46,47 This instability arises from symmetries of the lattice which result in points equidistant to four neighboring particles. Small perturbations break this symmetry, resulting in topological changes.…”

The transition from order to disorder in Voronoi Diagrams with applications

“…A second major generalisation modifies regular structures to produce non-Poissonian seeds for the Voronoi tessellation (NPVT). We select some methods, including perturbation of cubic structures Lucarini (2009), generation of seeds with controlled regularity Zhu et al (2014), an information geometric model to simulate graphene Dodson (2015), a 3D topological analysis Lazar et al (2015) and two-dimensional perturbed systems Leipold et al (2016).…”