We propose a class of goodness-of-fit tests for complete spatial randomness (CSR). In contrast to standard tests, our procedure utilizes a transformation of the data to a binary image, which is then characterized by geometric functionals. Under a suitable limiting regime, we derive the asymptotic distribution of the test statistics under the null hypothesis and almost sure limits under certain alternatives. The new tests are computationally efficient, and simulations show that they are strong competitors to other tests of CSR. The tests are applied to a real data set in gamma-ray astronomy, and immediate extensions are presented to encourage further work.

How do unique geometrical features of heterogeneous materials bring forth unique physical properties? Here, we study the link between the structure of sphere packings and a variety of physical properties using analytic approximations and rigorous bounds. In particular, we are interested in the most disordered among all mechanically stable packings of hard spheres (that are monodisperse and frictionless).

This maximally random jammed (MRJ) state exhibits an anomalous suppression of large-scale density fluctuations, known as hyperuniformity. Although the system is isotropic and locally disordered, it appears uniform on large scales. We compare the diffusion in the pore-space of the MRJ spheres to that of an equilibrium hard-sphere liquid or non-interacting spheres. Moreover, we study flow properties and effective conductivity; the latter also for anisotropic packings of spheroids. Therefore, we provide a comprehensive overview of rigorous bounds that connect these seemingly unrelated physical properties.

The most surprising result is found for electromagnetic waves that propagate through the MRJ sphere packings, where the wavelengths are much larger than the radii of the spheres. Usually disorder causes dissipation, but because of the unique property of hyperuniformity, the MRJ state forms, to a very good approximation, a dissipationless isotropic heterogeneous medium. This is demonstrated using an analytic strong-contrast expansion. It holds for any phase dielectric contrast ratio. The anomalous suppression of density fluctuations also suppresses the scattering of the electromagnetic waves.

This attribute could be useful for the design of photonic materials with novel structural color characteristics or color-sensing capabilities. Additive manufacturing fabrication techniques offer a simple production of samples for experiments with microwaves. So, our analytic results call for an experimental testing of these predicted qualitative trends in the physical properties associated with the MRJ structure.

We characterize the fluctuations in disordered systems of overlapping grains by calculating the second moments of a quite general class of robust shape descriptors. These include volume, surface area, Euler characteristic, and all other Minkowski functionals and tensors.

D. Hug, M. A. Klatt, G. Last, M. Schulte. Second order analysis of geometric functionals of Boolean models. In Lecture Notes in Mathematics Tensor Valuations and their Applications in Stochastic Geometry and Imaging (eds. E. B. Vedel Jensen and M. Kiderlen) 2177:339–383 (Springer International Publishing, 2017)

We especially focus on models of anisotropic heterogenous materials, that is, on anisotropic grain distributions.

How does the behaviour of the second moments and probability distributions differ for various models in finite systems for specific examples like aligned or isotropically
oriented rectangles? We derive explicit formulae, which are compared to the results of simulations. We show analytically that the asymptotic formulae for the thermodynamic limit are actually exact even for finite systems with periodic boundary conditions.

In the thermodynamic limit, the geometric functionals follow normal distributions, where the first and second moments are (explicitly) known, which could be used to construct hypothesis tests for random heterogeneous media.

Packings of hard, impenetrable spheres are useful models of granular media, low-temperature states of matter, suspensions and biological systems. What is the structure of the most disordered among all mechanical stable packings?

A unique property of this maximally random jammed (MRJ) state is that despite the local disorder, similar to a liquid, there is a hidden long-range order that anomalously suppresses density fluctuations on large length scales, more like in a crystalline solid. In a series of papers, we describe both the local and global structure of such disordered sphere packings using a variety of different structural characteristics.

In this second article, we derive explicit formulas but also apply Monte Carlo methods. By comparing the structure of MRJ packings to common models of disordered materials, our shape analysis helps to distinguish, despite seemingly similar features in all of those systems, their distinctly different structure.

Moreover, these structural characteristics are related to a host of different effective physical behavior, for example, flow or diffusion in these systems as well as their elastic moduli or electromagnetic properties. Our analysis thus links problems from material science, chemistry, physics, mathematics and biology.

The gyroid is an ordered network-like labyrinth bounded by minimal surfaces. It has become a house-hold name in soft materials with order on the nanometer scale, for example in the nanoporous photonic crystals of some green butterflies.

We here find by simulation that the same spatial gyroid structure forms spontaneously in nuclear matter at finite temperatures, as is prevalent in supernova explosions. While the structure of the gyroid in nuclear matter is the same as in soft materials, the length scale of a few femtometers is radically different, making this the discovery of the smallest reported gyroid found in dynamical simulations. The state of nuclear matter at this high nuclear density will greatly affect the neutrino transport during and after a supernova-explosion and is thus important to understand the production of heavy elements.

We characterize the structure of the maximally disordered packing among the set of all packings of monodisperse frictionless hard spheres, the so-called maximally random jammed (MRJ) sphere packing. Therefore, we compute the Minkowski functionals of the associated Voronoi cells and compare the structure to that of the Poisson point process (ideal gas) and of an equilibrium hard-sphere liquid.

In particular, we consider correlation functions or probability density functions of these Voronoi characteristics. Here we introduce and compute correlation functions and probability density functions of Minkowski functionals to quantify the global structure of the Voronoi diagram.

The local analysis using the distribution of the Voronoi volumes finds no qualitative difference for the structure of liquid or random jammed hard-sphere packings. In contrast to this, the higher-order statistical descriptors introduced here qualitatively distinguish the Voronoi structure of the MRJ sphere packings (prototypical glasses) from that of a hard-sphere liquid. We find strong anti-correlations in the MRJ sphere packings that arise because the MRJ state is “hyperuniform”.

We introduce a novel approach to source detection via structural deviations from typical features of a random homogeneous background. Minkowski functionals are powerful tools from integral geometry; in 2D, they are the area, the perimeter and the Euler characteristics (a topological quantity, which is given by the sum of all components minus the sum of all holes).

Via a combination of these different geometric measures that quantify the shape of level sets of a counts map, more information can be taken out of the same data without the need to assume prior knowledge about potential sources.

We introduce Minkowski sky maps that quantify local structural deviations. Moreover, they localize and visualize potential sources.