Keynote on a hidden order among disorder

It has been my great pleasure to open the 15th International Congress for Stereology and Image Analysis in Aarhus, Denmark, with my keynote on:

How to find a hidden order among disorder

where I gave an introduction and overview on hyperuniformity. The study of this anomalous suppression of density fluctuations on large length scales has shed light on a variety of seemingly unrelated fields, from the eyes of chicken to exotic many-particle ensembles and random matrices. The unique properties of hyperuniform amorphous materials (with a hidden order such that the system remains macroscopically uniform, despite not being crystalline) have recently led to intense research in physics, mathematics, material science, and biology. Aiming for an intuitive understanding of the rigorous mathematical definitions, I presented both basic concepts and recent examples.

I thank the organizers for a great conference in the last week of May 2019 with many stimulating discussions across disciplines.


Focus Session at the APS March Meeting 2019

It has been my great pleasure to organize together with Lisa Manning, Gregory Grason, and Gerd Schröder-Turk a GSOFT Focus Session at the APS March Meeting in Boston this year:

"Hyperuniformity and Optimal Tessellations: Structure, Formation and Properties"

After recent breakthroughs in the search for ordered optimal tessellations (for example, including Frank-Kasper phases in copolymer melts), now findings of the optimal properties of amorphous tessellations are emerging, e.g., in biological tissues.

At the same time, there have been intensive studies of amorphous systems with an anomalous suppression of density fluctuations on large length scales, known as hyperuniformity. This geometric concept qualitatively and quantitatively characterizes a hidden-order in amorphous states that allows for unique physical properties – combining those of crystalline and disordered phases. Thus it offers candidates for optimal amorphous tessellations of space.

The two invited speakers of our session on Thursday, March 7, were Jasna Brujic and Salvatore Torquato.


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.