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Shape of Random Structures in Nature: from Supernovae to Medical Diagnostics

Each year two speakers are chosen to talk about their research at the Bavarian evening. Michael Klatt had the chance to present his doctoral research on the shape of random structures in nature:

"From the large-scale structure of the universe to pasta shapes in nuclear matter: random spatial structures appear ubiquitously on nearly all length scales in very di fferent physical, chemical, or biological systems, as well as, in food industry, geology, medicine, astronomy or cosmology. The morphology varies from cellular structures, packings of grains, or point processes to random fields. In systems with a complex structure, there is a close interconnection of physics and geometry, and physical insight is often best achieved by a rigorous characterization of the structure. However, the morphology of disordered systems is a remarkably incoherent concept, and in order to characterize these complex shapes, advanced mathematical tools are needed, i.e., sensitive and robust measures of structure. Modern imaging techniques, like confocal microscopy, electron or X-ray tomography, or high resolution magnetic resonance imaging, can resolve fine structural details in real space and call for sensitive real space measures of shape.

Aiming for both rigorous mathematical foundation and relevance to the application, I use integral geometric measures for a sensitive and comprehensive morphometric analysis, the so-called Minkowski functionals and tensors, extending the notion of volume and surface area to scalar and tensorial morphometric measures. I apply these versatile shape characteristics for a refi ned morphometric analysis of a variety of systems on very di fferent length-scales from nuclear matter over condensed and soft matter to gamma-ray astronomy, also including analytic calculations for the most common models of disordered structures. The structures of interest range from maximally disordered confi gurations to spontaneously forming regular structures. For a thorough structure characterization, I do not only determine the average shape measures, but their variance or even their complete probability distribution, e.g., to detect statistically significant inhomogeneities in random noise. Moreover, I put a special emphasis on the quanti fication of anisotropy.

So far, mainly ad-hoc solutions are used as anisotropy measures which are explicitly designed for specific systems or they are restricted to special geometric features, e.g., to interfacial anisotropy, like the mean intercept length (MIL) tensor which is a common measure of the anisotropy of trabecular bone. In analytic and numeric calculations for a standard model of porous media, i.e., overlapping grains which are randomly distributed in space, we show that the MIL tensor can appear perfectly isotropic for structures which are obviously anisotropic and the common approach can sometimes be even ill-defined.

In contrast to the ad-hoc solutions, the Minkowski tensors allow for a comprehensive and systematic approach to quantify various aspects of structural anisotropy. We have both analytically and numerically determined the Minkowski tensors of common models of heterogeneous materials, including overlapping random grains, for which we have provided an accurate structure characterization. We have shown that the Minkowski tensors clearly detect anisotropy for systems which appear isotropic w.r.t. the common MIL analysis. Therefore, the Minkowski tensors are a promising tool for a diagnosis based on an anisotropy analysis, e.g., of osteoporosis, which causes not only a loss of bone material but also a deterioration of the microarchitecture of trabecular bone.

The model of overlapping grains with preferred orientation is then used to study the geometry and topology of a system spanning cluster of connected components, the so-called percolating cluster. For example, I investigate how the topology depends on the anisotropy of the system. The percolation threshold, i.e., the volume fraction at which such a percolating cluster appears, depends on the anisotropy of the system, and the Minkowski functionals allow for explicit estimates of this threshold. However, even the most anisotropic model will simultaneously percolate in all directions, i.e., the percolation threshold is isotropic, which is linked to the uniqueness of the percolating cluster.

To analyze the global structure also of cellular systems, the local analysis based on single cell characteristics is extended by introducing global correlation functions of Minkowski functionals,
which find for tessellations with qualitatively very similar local structure a distinctly different
global morphology. Moreover, these Minkowski correlation functions can distinguish systems
with equal standard two-point correlation functions.

Even on the smallest length scale, we use the Minkowski functionals to characterize and classify simulated exotic states of nuclear matter, so-called nuclear pasta, which are expected to appear, e.g., in supernova explosions. We show how nuclear pasta and butterfly scales can be united by the same spatial microstructure, the gyroid, which 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. With the help of the Minkowski functionals, we find among spontaneously forming random labyrinthine pasta shapes the same spatial gyroid structure. However, 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.

Finally, we use the Minkowski functionals for a morphometric data analysis in gamma-ray astronomy: by characterizing the morphology of a noisy sky-map more information can be taken out of the same data without assuming prior knowledge about the source. I derive an accurate estimate of the distribution of the background structure, which is simultaneously characterized by all Minkowski functionals. Such a refined structure characterization can eventually detect formerly undetected sources.

The Minkowski functionals and tensors allow for a versatile morphometric analysis which can sensitively and comprehensively analyze very di erent complex structures on all length scales and can thus provide physical insight via shape characterization."