PhD thesis – Michael Klatt

Abstract

From the large-scale structure of the universe to pasta shapes in nuclear matter: random or disordered spatial structures appear ubiquitously on nearly all length scales in very different physical, chemical, or biological systems, as well as, in food industry, geology, medicine, astronomy or cosmology. The shapes vary from cellular structures, packings of grains, or point processes to random fields. In systems with 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. Because physics is a quantitative science, these relations can only be understood by a quantitative shape analysis. However, the structure of disordered systems is a remarkably incoherent concept; in order to characterize all of these complex shapes with the same measures, advanced mathematical tools are needed, i.e., versatile, sensitive, and robust measures of structure. Aiming for both rigorous mathematical foundation and relevance to the application, I use integral geometric measures for a sensitive and comprehensive morphometric analysis. These measures extend the notion of volume and surface area to scalar and tensorial morphometric measures, the so-called Minkowski functionals and tensors.

In this thesis, I use and extend these integral geometric measures to characterize the structures of a variety of systems on very different length scales ranging from nuclear matter over condensed and soft matter to gamma-ray astronomy, including both simulations and experimental data, as well as, analytic calculations for common and important models of stochastic geometry. The shapes of interest range from maximally disordered configurations to systems forming spontaneously regular structures. The same shape descriptors are applied to all of these systems, providing physical insight via a characterization of the complex geometry. A special emphasis is on the extension of the morphometric analysis to higher moments of the structure distributions or even to the full probability distributions, as well as, to anisotropic random spatial structures. In the latter case, I use tensorial shape descriptors to characterize the anisotropy, where I also investigate the information content of higher-rank tensors.

I thoroughly characterize the anisotropy of common models of porous media, overlapping grains and Gaussian random fields. I compare anisotropy measures w.r.t. their sensitivity and show how the Minkowski tensors resolve the disadvantages of a common measure of anisotropy. Explicit formulas for the mean values of the anisotropy measures are derived and compared to simulations. For the Minkowski functionals of overlapping grains, I also study the covariance structure and a central limit theorem. For the Gaussian random field, the tensors of higher rank are shown to contain additional anisotropy information as compared to the tensor of rank two. However, surprisingly, the latter is nevertheless sufficient to estimate all model parameters that are necessary to determine all Minkowski tensors of arbitrary rank which characterize the interfacial anisotropy of the level sets of the Gaussian random field. This relation could be used to test for non-Gaussianities in anisotropic random fields. The results on the models are important for applications, e.g., to estimate parameters, for null hypothesis tests, or to adjust the model to experimental data, and they results provide fundamental insights, e.g., how geometric and topological characteristics either depend on the specific systems parameters or when they instead exhibit a universal behavior. The models of overlapping grains with preferred orientation are then used to study the topology of system spanning clusters, so-called percolating clusters; especially the dependence of their topology on the anisotropy of the system. Their topology can, e.g., be connected to transport properties. The percolation threshold, i.e., the volume fraction at which such a percolating cluster appears, depends on the anisotropy of the system. However, interestingly, even the most anisotropic model simultaneously percolates in all directions. In other words, the percolation threshold is isotropic, which is linked to the uniqueness of the percolating cluster. The Minkowski functionals allow for explicit estimates of the threshold. To analyze both the local and the global structure of cellular systems, I extend an analysis which uses local cell characteristics by introducing global correlation functions. I find for tessellations with qualitatively nearly indistinguishable local structure distributions a significantly different global structure, which here arises from disordered hyperuniformity, which is considered to be a new state of matter. Moreover, I show that the here defined Minkowski correlation functions are supreme to standard two-point correlation functions in that they incorporate higher $n$-point information which could be used, e.g., for sensitive hypothesis tests to distinguish models of galaxy distributions. Even on an extremely small length scale, I use the Minkowski functionals to characterize and classify simulated exotic states of nuclear matter, so-called nuclear pasta, which is expected to appear, e.g., in supernova explosions. Interestingly, among these pasta shapes a spontaneously formed gyroid is identified, a special minimal surface which also appears many orders of magnitudes larger in biology, making this the discovery of the smallest reported gyroid found in dynamical simulations and demonstrating the universal principles of shape formation. The binding energy of the pasta matter seems to be closely related to the Minkowski functionals, but anisotropic deformations, which can be quantified by Minkowski tensors, seem to be irrelevant. Finally, I use the Minkowski functionals for a sensitive morphometric data analysis in gamma-astronomy: via a combination of different geometric measures more information can be taken out of the same data without the need to assume prior knowledge about potential sources. I improve the method, especially the correction of detector effects and via detailed simulations to test the significance of structural deviations. Most importantly, I derive an accurate estimate of the distribution of background structure jointly characterized by all Minkowski functionals for large scan windows. Thus, I am able to gain an increase in sensitivity for the morphometric analysis: in the same data formerly undetected sources can eventually be detected. I then apply the method also to H.E.S.S. sky maps. Moreover, I present an alternative approach using a less precise knowledge of the background structure, for which I show how the joint characterization by all three Minkowski functionals increases the sensitivity to detect inhomogeneities in examples where there is no excess in the total number of counts.

In conclusion, the Minkowski functionals and tensors allow for an intuitive and versatile morphometric analysis that can sensitively and comprehensively characterize the geometry of very different complex structures on nearly all length scales providing physical insight.