Abstracts

Abstract: 

Hyperuniformity refers to the suppression of density fluctuations at large scales. Its classical definition utilizes information at the largest accessible scales, such as the scaling of the number variance for large sampling window size or the structure factor at small wave numbers. At the same time, hyperuniform (HU) configurations have distinctive local characteristics: strong HU characters can even be guessed by the naked eye of an expert. More importantly, such characteristics are expected to affect properties like response to deformation and filtration characteristics, and it is thus of general interest to tailor them for various applications. This presentation illustrates how local features in HU configurations can be characterized via their topological properties. In particular, we focus on point clouds and characterize their persistent homology as well as the statistics of local graph neighborhoods. We then discuss implications related to this novel information. We find that varying the structure factor results in configurations with systematically different topological properties. Moreover, these topological properties are almost conserved for subsets of HU point clouds, establishing a connection between finite-sized systems and idealized reference arrangements. As with classical measures, HU arrangements result in a distinct class of patterns realizing an order-to-disorder transition also when looking at local topological neighborhoods. Finally, concepts for inverse design and detection of HU patterns leveraging persistent homology are discussed together with future perspectives and potential applications.

 

Title: Hyperuniformity of dependent perturbed lattices.

 

Abstract: We ask whether a stationary lattice in dimension $d$ whose points are shifted by identically distributed but possibly dependent perturbations remains hyperuniform. When $d = 1$ or $2$, we show that it is the case when the perturbations have a finite $d$-moment, and that this condition is sharp. When $d \geq 3$, we construct arbitrarily small perturbations such that the resulting point process is not hyperuniform.

Joint work with: Daniela Flimel, Martin Huesmann and Thomas Leblé

 

Daniela Flimmel

Title: Fitting regular point patterns with hyperuniform perturbed lattices

Abstract: When detecting regularity within a dataset, Gibbsian models became a common choice. It has been shown that a large class of repulsive potentials lead to non-hyperuniform Gibbs point processes. On the other hand, perturbed lattices could preserve their natural repulsivity and therefore, should be considered a valid candidate for the model as well. It is nowadays understood that perturbed lattices form a wide inventory of point processes from class I hyperuniform to above-Poisson fluctuating. Depending on the variability in the dataset, one can choose a suitable perturbation and expect straightforward intuition in the model as well as fast computation times. Selected results will be presented on the grain centers of nitinol alloy in 3D.

 

Martin Huesmann

Title: Hyperuniformity, Coulomb Energy and Wasserstein distance

Abstract: We show an interplay between the three properties of two-dimensional point processes: A) HU, B) Finite regularized Coulomb energy, and C) Finite 2-Wasserstein distance to Lebesgue. In particular, we will show that B) implies C) implies A) and there are counterexamples for the converse directions.

This is based on joint work with Thomas Leblé.

 

Mathias Kasiulis

Title: Fast generation of large hyperuniform point patterns

Abstract: In physics, recent work has highlighted the potential of correlated disordered materials (aperiodic non-Poissonian structures) to achieve functions. For instance, in photonics, such structures may display properties usually associated with periodic media (transparency, birefringence, structural coloration), and even give rise to unique phenomena like Anderson localization. The majority of existing results on correlated disordered media address only a few classes of disorder, e.g. stealthy hyperuniform systems. Furthermore, they are largely limited to small systems, consisting of only a few hundreds or thousands of particles. Both constraints stem from limitations in the numerical algorithms used to generate correlated disordered structures.