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Hidden order in disorder

Hidden order in disorder

Hyperuniformity is a geometric concept to probabilistically characterise the structure of ordered and disordered materials. For example, all perfect crystals, perfect quasicrystals, and special disordered systems are hyperuniform.

The study of how large structures are partitioning space into cells with specific extreme geometrical features is a crucial topic in many disciplines of science and technology. Researchers from Karlsruhe Institute of Technology (KIT) and colleagues from other countries have discovered that in amorphous, or disordered, systems, optimizing the moment of inertia of individual cells gradually results in the same structure, despite the fact that it remains amorphous.

The team analyzes the Quantizer problem, defined as the optimization of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that pack as closely as possible. Scientists have been studying the optimal tessellation of three-dimensional space for a long time, being relevant to a variety of practical applications, including telecommunications, image processing, and complicated granules, among others.

The goal is to partition space into cells, and all points in a cell to be located as closely as possible to the cell center, intuitively speaking.

Dr. Michael Andreas Klatt
Schematic representation of a Lloyd iteration. Lloyd iterations convert an initial point set to a point set with a centroidal Voronoi diagram. In each iteration, the algorithm first computes the Voronoi cells for all points. Then each point (black circle) is replaced by the centre of mass (yellow cross) of its Voronoi cell.

The team deployed the Lloyd algorithm, a method to partition space into uniform regions. Every area has a unique center and contains the points in space that are closest to it. Voronoi cells are the name given to such areas. The Voronoi diagram is made up of all points that have more than one nearest center, establishing the region boundaries. They discovered that all entirely amorphous, i.e. disordered, states not only remain completely amorphous, but that the initially different processes converge to a statistically indistinguishable ensemble by investigating stepwise local optimization of distinct point patterns. Stepwise local optimization also compensates for extreme global density changes quickly.

The resulting structure is nearly hyperuniform. It does not exhibit any obvious, but a hidden order on large scales.

Convergence of Lloyd’s algorithm in 3D

The findings show an unexpected universality for the complex interactions of the Quantizer energy in a detailed structure analysis of its local minima. The existence of universal effectively hyperuniform and fully amorphous states are the converged solutions of Lloyd’s algorithm in 3D.

Even when the initial configuration is hyperfluctuating, where the structure factor diverges for small wavenumbers, the system quickly becomes under Lloyd iterations effectively hyperuniform. This demonstrates the strong suppression of density fluctuations on large but finite length scales that is consistent with effective hyperuniformity.

See Also

This hidden order in the studied amorphous systems is proved to be universal, i.e. stable and independent of properties of the initial state, providing the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties.

Universal hidden order in amorphous cellular geometries, Michael A. Klatt, Jakov Lovrić, Duyu Chen, Sebastian C. Kapfer, Fabian M. Schaller, Philipp W. A. Schönhöfer, Bruce S. Gardiner, Ana-Sunčana Smith, Gerd E. Schröder-Turk & Salvatore Torquato

Published: February 2019

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