Topological measures of order for imperfect two-dimensional Bravais lattices

Motivated by patterns with defects in natural and laboratory systems, we develop two quantitative measures of order for imperfect Bravais lattices in the plane. A tool from topological data analysis called persistent homology combined with the sliced Wasserstein distance, a metric on point distributions, are the key components for defining these measures. The measures generalize previous measures of order using persistent homology that were applicable only to imperfect hexagonal lattices in two dimensions. We illustrate the sensitivities of these measures to the degree of perturbation of perfect hexagonal, square, and rhombic Bravais lattices. We also study imperfect hexagonal, square, and rhombic lattices produced by numerical simulations of pattern-forming partial differential equations. These numerical experiments serve to compare the measures of lattice order and reveal differences in the evolution of the patterns in various partial differential equations.

Automated summary

Motivated by patterns with defects, this study develops two quantitative measures of order for imperfect Bravais lattices in the plane. Persistent homology and the sliced Wasserstein distance are used to define these measures. These measures generalize previous measures of order and are sensitive to the degree of perturbation in various lattice types and pattern-forming partial differential equations.