Low-temperature statistical mechanics of the Quantizer problem

Following up on our finding of a universal hidden order in tessellations with the Quantizer energy functionals (where the cells minimize the second moment of the volume distribution) using Lloyd’s algorithm, we now compared these states with quenches from equilibrium states at finite temperature.

Upon sufficiently fast non-equilibrium cooling, the system adopts similar states as the effectively hyperuniform inherent structures found by Lloyd’s algorithm and prevents the ordering transition into a body-centered cubic ordered structure.


Hyperuniform and rigid stable matchings

With respect to the degree of order and disorder, a snapshot of the ideal gas with completely independent particle positions (in mathematics known as the realization of a Poisson point process) is the exact opposite of a lattice with a perfect short- and long-range order.

We have defined a stable matching between these two extremes and proved that the resulting pattern can locally be virtually indistinguishable from the ideal gas but at large scales it inherits two remarkable global properties of the lattice: hyperuniformity and number rigidity, as demonstrated in our supplementary video.

M. A. Klatt, G. Last, D. Yogeshwaran. Hyperuniform and rigid stable matchings. Random Struct Alg. 57:439–473 (2020)

In the stable matching, points of the lattice and the ideal gas prefer to be close to each other. Assuming that there are on average more points per unit area in the ideal gas than in the lattice, the matched points in the ideal gas form a new point process that inherits properties from both the lattice and the ideal gas.

In fact, if the mean number of points per unit area converge, the new point pattern becomes almost indistinguishable from the ideal gas in any finite observation window, while its large-scale density fluctuations remain anomalously suppressed similar to the lattice.

We prove that the resulting point process is hyperuniform and number rigidity also for a stable matching between a lattice and not only the ideal gas but a class of correlated point processes, known as determinantal point processes. For a one-sided matching in 1D, our proof applies to quite general point processes.

Our paper solves an open question from the seminal paper by Salvatore Torquato and Frank Stillinger that introduced the notion of hyperuniformity. Our matched point processes generalizes a 1D hyperuniform point process by Goldstein, Lebowitz, Speer.

You can easily match two arbitrary point patterns using my R-package: matchingpp


Geometric Functionals of Fractal Percolation

Fractal percolation exhibits a dramatic topological phase transition, changing abruptly
from a dust-like set to a system-spanning cluster. Where this transition occurs is unknown
and notoriously difficult to estimate. In continuum percolation, the percolation thresholds have been approximated well using Minkowski functionals. Motivated by the question of whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process.

M. A. Klatt and S. Winter. Geometric functionals of fractal percolation. Adv. Appl. Probab. 52:1085–1126 (2020)

We establish the existence of the limit functionals and obtain explicit formulas for them as well as for their finite approximations (in excellent agreement with simulation result). They provide new geometric insights into fractal percolation. For example, we provide a mathematically rigorous notion of fractal subdimensions that have been previously defined in physics as an approach to refined information about fractal sets beyond fractal dimension.

My code to simulate fractal percolation and to analyze our geometric functionals is freely available at GitHub.


Detecting structured sources in noisy images via Minkowski maps

Distinguishing a feature from background noise is often the key to discovering physical phenomena in spatial data for a wide range of applications. We show how the sensitivity of a null hypothesis test can be dramatically increased by additional geometric information without the need for any prior knowledge about potential sources.

M. A. Klatt and K. Mecke. Detecting structured sources in noisy images via Minkowski maps. EPL (Europhysics Letters) 128:60001-p1–8 (2019)

We present a morphometric analysis technique, based on Minkowski functionals, to quantify the shape of structural deviations in greyscale images. It identifies important features in noisy spatial data, especially for short observation times and low statistics.
Without assuming any prior knowledge about potential sources, the additional shape information can increase the sensitivity by 14 orders of magnitude compared to previous methods. Rejection rates can increase by an order of magnitude.

As a key ingredient to such a dramatic increase, we accurately describe the distribution of the homogeneous background noise in terms of the density of states for the area, perimeter, and Euler characteristic of random black-and-white images. Remarkably, the density of states can vary by 64 orders of magnitude. We apply our technique to data of the H.E.S.S. experiment.

The density of states and my code for Minkowski maps are freely available at GitHub.

All simulated data and parameters underlying our study are available at Zenodo.


Cloaking the underlying long-range order of randomly perturbed lattices

How can we cloak the Bragg peaks of a lattice with random perturbations and will traces of the underlying long-range order persist nevertheless? Our affirmative answer to both questions has been published as an Editors’ Suggestion:

M. A. Klatt, J. Kim, S. Torquato. Cloaking the underlying long-range order of randomly perturbed lattices. Phys. Rev. E 101, 032118-1–9 (2020)

The Editors of Physical Review E have suggested our paper as a Highlight:
“When a regular lattice is perturbed, traces of the original lattice usually remain visible as Bragg peaks in the diffraction pattern. The authors of this paper consider uniformly randomized lattices and show that fine-tuned distributions of perturbations can hide the Bragg peaks. Interestingly, as the strength of the perturbations increases, long-range order oscillates and Bragg peaks appear and disappear.”


Phoamtonics: Photonic band gaps of 3D foams

In our paper, published in PNAS, we show how to construct a foam-based photonic crystal with a substantial band gap using the famous Weaire-Phelan foam.

M. A. Klatt, P. J. Steinhardt, S. Torquato. Phoamtonic designs yield sizeable 3D photonic band gaps. Proc. Natl. Acad. Sci. U.S.A. 116:23480–23486 (2019)

A report by Steven Schultz about our study has been featured on the main webpage of Princeton University:


See also, the original post of the engineering department:

The Weaire-Phelan foam has the smallest surface area among all known tessellations with equal-volume cells. While it has only a slightly smaller surface area than the Kelvin foam. Its corresponding photonic network provides a distinctly larger band gap and a much smaller critical refractive index. To determine the latter, we have created a detailed map of gap sizes.

Among the promising prospects that phoamtonics offers for applications are multifunctional characteristics, the self-organization of large photonic networks, and a high degree of isotropy.

We published all data generated or analyzed for our phoamtonics study, including configurations, parameter files, raw output, and postprocessed data, in a Zenodo
( DOI: 10.5281/zenodo.3401635


Strength of bacterial adhesion on nanostructured surfaces quantified by substrate morphometry

In this paper, we combine physics, microbiology and mathematics to study how bacteria attach to nanostructured surfaces, and how topography of nanostructured substrates can directly quantify the adhesion.

C. Spengler, F. Nolle, J. Mischo, T. Faidt, S. Grandthyll, N. Thewes, P. Jung, M.
Koch, F. Müller, M. Bischoff, M. A. Klatt, K. Jacobs. Strength of bacterial adhesion on nanostructured surfaces quantified by substrate morphometry. Nanoscale 11:19713–19722 (2019)

Aiming for 3D-structured materials for or against bio-adhesion, we studied mechanisms by which the bacteria adhere to different types of materials. Our insights help, for example, to prevent adhesion of pathogens (like Staphylococcus aureus) before the bacteria can begin to grow a biofilm, which protects individual bacteria from attack by antibiotics.

We found that the measured forces with which the individual bacterial cells adhered to nanostructured surfaces decrease with an increasing size of these nanostructures. In fact, with our morphometric analysis based on Minkowski functionals, we found that the strength of the adhesion force can be predicted from measurements of the surface area that is accessible to the tethering proteins with which the bacteria adhere to the surface. More precisely, we found that the ratio of the experimentally measured adhesion forces at the rough surface compared to a smooth surface agrees with the fraction of surface area that is accessible to the tethering proteins, which are about 50 nm long.

There are press releases in English and German.


Universal hidden order in amorphous cellular geometries

In this paper, we analyze the evolution of random point patterns and their Voronoi diagrams as “quantizer” of space. Applying an iterative local optimization of their so-called Quantizer energy, we show that the patterns converge to the apparently same effectively hyperuniform state regardless of their initial conditions.

M.A. Klatt, J. Lovrić, D. Chen, S. C. Kapfer, F. M. Schaller, P. W. A. Schönhöfer, B. S. Gardiner, A.-S. Smith, G. E. Schröder-Turk, S. Torquato. Universal hidden order in amorphous cellular geometries. Nature Communications 10, 811 (2019)

Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. Here we investigate the Quantizer problem, defined as the optimisation of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that tile space are preferred. We employ Lloyd’s centroidal Voronoi diagram algorithm to solve this problem and find that it converges to disordered states associated with deep local minima. These states are universal in the sense that their structure factors are characterised by a complete independence of a wide class of initial conditions they evolved from. They moreover exhibit an anomalous suppression of long-wavelength density fluctuations and quickly become effectively hyperuniform. Our findings warrant the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties.


Goodness-of-fit tests for complete spatial randomness based on Minkowski functionals of binary images

In this paper we use the Minkowski functionals for a rigorous null hypothesis test for complete spatial randomness.

Excursion set of a binned Poisson point process

B. Ebner, N. Henze, M. A. Klatt, K. Mecke. Goodness-of-fit tests for complete spatial randomness based on Minkowski functionals of binary images. Electron. J. Statist 12:2873–2904 (2018)

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.


Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

In this paper, we study the physical properties of the hyperuniform maximally random jammed sphere packings.

Visualization of electromagnetic waves propagating through an MRJ sphere packing

M. A. Klatt and S. Torquato. Characterization of maximally random jammed sphere packings. III. Transport and Electromagnetic Properties Via Correlation Functions. Phys. Rev. E, 97:012118-1–17 (2018)

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.