arXiv subjects

Klaus Mecke

Publications and source records attributed to Klaus Mecke.

At least 19 recordsLinked to original sources

arXiv2021

Characterization of Anisotropic Gaussian Random Fields by Minkowski Tensors

Gaussian random fields are among the most important models of amorphous spatial structures and appear across length scales in a variety of physical, biological, and geological applications, from composite materials to geospatial data. Anisotropy in such systems can sensitively and comprehensively be characterized by the so-called Minkowski tensors from integral geometry. Here, we analytically calculate the expected Minkowski tensors of arbitrary rank for the level sets of Gaussian random fields. The explicit expressions for interfacial Minkowski tensors are confirmed in detailed simulations. We demonstrate how the Minkowski tensors detect and characterize the anisotropy of the level sets, and we clarify which shape information is contained in the Minkowski tensors of different rank. Using an irreducible representation of the Minkowski tensors in the Euclidean plane, we show that higher-rank tensors indeed contain additional anisotropy information compared to a rank two tensor. Surprisingly, we can nevertheless predict this information from the second-rank tensor if we assume that the random field is Gaussian. This relation between tensors of different rank is independent of the details of the model. It is, therefore, useful for a null hypothesis test that detects non-Gaussianities in anisotropic random fields.

cond-mat.dis-nn↗

arXiv2021

Detecting structured sources in noisy images via Minkowski maps

Astronomy, biophysics, and material science often depend on the possibility to extract information out of faint spatial signals. Here we present a morphometric analysis technique 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 $Ω(A,P,χ)$ for the area $A$, perimeter $P$, and Euler characteristic $χ$ of random black-and-white images. The technique is successfully applied to data of the H.E.S.S. experiment for the detection of faint extended sources.

physics.data-an↗

arXiv2021

Tracking down the origin of superbubbles and supergiant shells in the Magellanic Clouds with Minkowski tensor analysis

We develop an automatic bubble-recognition routine based on Minkowski functionals (MF) and tensors (MT) to detect bubble-like interstellar structures in optical emission line images. Minkowski functionals and MT are powerful mathematical tools for parameterizing the shapes of bodies. Using the papaya2-library, we created maps of the desired MF or MT of structures at a given window size. We used maps of the irreducible MT $ψ_2$, which is sensitive to elongation, to find filamentary regions in H$α$, [SII], and [OIII] images of the Magellanic Cloud Emission Line Survey (MCELS). Using the phase of $ψ_2$, we were able to draw lines perpendicular to each filament and thus obtain line-density maps. This allowed us to find the center of a bubble-like structure and to detect structures at different window sizes. The detected bubbles in all bands are spatially correlated to the distribution of massive stars, showing that we indeed detect interstellar bubbles without large spatial bias. Eighteen out of 59 supernova remnants in the Large Magellanic Cloud (LMC) and 13 out of 20 superbubbles are detected in at least one wavelength. The lack of detection is mostly due to surrounding emission that disturbs the detection, a too small size, or the lack of a (circular) counterpart in our emission line images. In line-density maps at larger scales, maxima can be found in regions with high star formation in the past, often inside supergiant shells (SGS). In SGS LMC 2, there is a maximum west of the shell where a collision of large gas clouds is thought to have occurred. In the Small Magellanic Cloud (SMC), bubble detection is impaired by the more complex projected structure of the galaxy. Line maps at large scales show large filaments in the SMC in a north-south direction, especially in the [SII] image. The origin of these filaments is unknown.

astro-ph.GA↗

arXiv2017

Morphometric analysis in gamma-ray astronomy using Minkowski functionals: II. Joint structure quantification

We pursue a novel morphometric analysis to detect sources in very-high-energy gamma-ray counts maps by structural deviations from the background noise. Because the Minkowski functionals from integral geometry quantify the shape of the counts map itself, the morphometric analysis includes unbiased structure information without prior knowledge about the source. Their distribution provides access to intricate geometric information about the background. We combine techniques from stochastic geometry and statistical physics to determine the joint distribution of all Minkowski functionals. We achieve an accurate characterization of the background structure for large scan windows (with up to $15\times15$ pixels), where the number of microstates varies over up to 64 orders of magnitude. Moreover, in a detailed simulation study, we confirm the statistical significance of features in the background noise and discuss how to correct for trial effects. We also present a local correction of detector effects that can considerably enhance the sensitivity of the analysis. In the third paper of this series, we will use the here derived refined structure characterization for a more sensitive data analysis that can detect formerly undetected sources.

astro-ph.IM↗

arXiv2017

Morphometric analysis in gamma-ray astronomy using Minkowski functionals: III. Sensitivity increase via a refined structure quantification

We pursue a novel morphometric analysis to detect sources in very-high-energy gamma-ray counts maps by structural deviations from the background noise without assuming any prior knowledge about potential sources. The rich and complex structure of the background noise is characterized by Minkowski functionals from integral geometry. By extracting more information out of the same data, we aim for an increased sensitivity. In the first two papers, we derived accurate estimates of the joint distribution of all Minkowski functionals. Here, we use this detailed structure characterization to detect structural deviations from the background noise in a null hypothesis test. We compare the analysis of the same simulated data with either a single or all Minkowski functionals. The joint structure quantification can detect formerly undetected sources. We show how the additional shape information leads to the increase in sensitivity. We explain the very unique concepts and possibilites of our analysis compared to a standard counting method in gamma-ray astronomy, and we present in an outlook further improvements especially for the detection of diffuse background radiation and generalizations of our technique.

astro-ph.IM↗

arXiv2017

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

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.

stat.ME↗

arXiv2017

Anisotropy in finite continuum percolation: Threshold estimation by Minkowski functionals

We examine the interplay between anisotropy and percolation, i.e., the spontaneous formation of a system spanning cluster in an anisotropic model. We simulate an extension of a benchmark model of continuum percolation, the Boolean model, which is formed by overlapping grains. Here we introduce an orientation bias of the grains that controls the degree of anisotropy of the generated patterns. We analyze in the Euclidean plane the percolation thresholds above which percolating clusters in $x$- and in $y$-direction emerge. Only in finite systems, distinct differences between effective percolation thresholds for different directions appear. If extrapolated to infinite system sizes, these differences vanish independent of the details of the model. In the infinite system, the uniqueness of the percolating cluster guarantees a unique percolation threshold. While percolation is isotropic even for anisotropic processes, the value of the percolation threshold depends on the model parameters, which we explore by simulating a score of models with varying degree of anisotropy. To which precision can we predict the percolation threshold without simulations? We discuss analytic formulas for approximations (based on the excluded area or the Euler characteristic) and compare them to our simulation results. Empirical parameters from similar systems allow for accurate predictions of the percolation thresholds (with deviations of $<5\%$ in our examples), but even without any empirical parameters, the explicit approximations from integral geometry provide, at least for the systems studied here, lower bounds that capture well the qualitative dependence of the percolation threshold on the system parameters (with deviations of $5\%$--$30\%$). As an outlook, we suggest further candidates for explicit and geometric approximations based on second moments of the so-called Minkowski functionals.

cond-mat.dis-nn↗

arXiv2016

Cell shape analysis of random tessellations based on Minkowski tensors

To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic models. In the context of image analysis of synthetic and biological materials, this question is central to the problem of inferring information about formation processes from spatial measurements of resulting random structures. We address this question by a theory-based simulation study of shape indices derived from Minkowski tensors for a variety of tessellation models. We focus on the relationship between two indices: an isoperimetric ratio of the empirical averages of cell volume and area and the cell elongation quantified by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, and Gibbs hard-core and random sequential absorption processes as well as for Laguerre tessellations of polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data are complemented by mechanically stable crystalline sphere and disordered ellipsoid packings and area-minimising foam models. We find that shape indices of individual cells are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of the shape indices between many of these tessellation models. Given a realization of a tessellation, these shape indices can narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by density-resolved volume-shape correlations.

cond-mat.soft↗

arXiv2016

Spectral properties of unimodular lattice triangulations

Random unimodular lattice triangulations have been recently used as an embedded random graph model, which exhibit a crossover behaviour between an ordered, large-world and a disordered, small-world behaviour. Using the ergodic Pachner flips that transform such triangulations into another and an energy functional that corresponds to the degree distribution variance, Markov chain Monte-Carlo simulations can be applied to study these graphs. Here, we consider the spectra of the adja cency and the Laplacian matrix as well as the algebraic connectivity and the spectral radius. Power law dependencies on the system size can clearly be identified and compared to analytical solutions for periodic ground states. For random triangulations we find a qualitative agreement of the spectral properties with well-known random graph models. In the microcanonical ensemble analytical approximations agree with numerical simulations. In the canonical ensemble a crossover behaviour can be found for the algebraic connectivity and the spectral radius, thus combining large-world and small-world behavior in one model. The considered spectral properties can be applied to transport problems on triangulation graphs and the crossover behaviour allows a tuning of important transport quantities.

cond-mat.dis-nn↗

arXiv2016

Genus dependence of the number of (non-)orientable surface triangulations

Topological triangulations of orientable and non-orientable surfaces with arbitrary genus have important applications in quantum geometry, graph theory and statistical physics. However, until now only the asymptotics for 2-spheres are known analytically, and exact counts of triangulations are only available for both small genus and small triangulations. We apply the Wang-Landau algorithm to calculate the number $N(m,h)$ of triangulations for several order of magnitudes in system size $m$ and genus $h$. We verify that the limit of the entropy density of triangulations is independent of genus and orientability and are able to determine the next-to-leading and the next-to-next-to-leading order terms. We conjecture for the number of surface triangulations the asymptotic behavior \begin{equation*} N(m,h) \rightarrow (170.4 \pm 15.1)^h m^{-2(h - 1)/5} \left( \frac{256}{27} \right)^{m / 2}\;, \end{equation*} what might guide a mathematicians proof for the exact asymptotics.

physics.comp-ph↗

arXiv2015

Non-universal Voronoi cell shapes in amorphous ellipsoid packings

In particulate systems with short-range interactions, such as granular matter or simple fluids, local structure plays a pivotal role in determining the macroscopic physical properties. Here, we analyse local structure metrics derived from the Voronoi diagram of configurations of oblate ellipsoids, for various aspect ratios $α$ and global volume fractions $ϕ_g$. We focus on jammed static configurations of frictional ellipsoids, obtained by tomographic imaging and by discrete element method simulations. In particular, we consider the local packing fraction $ϕ_l$, defined as the particle's volume divided by its Voronoi cell volume. We find that the probability $P(ϕ_l)$ for a Voronoi cell to have a given local packing fraction shows the same scaling behaviour as function of $ϕ_g$ as observed for random sphere packs. Surprisingly, this scaling behaviour is further found to be independent of the particle aspect ratio. By contrast, the typical Voronoi cell shape, quantified by the Minkowski tensor anisotropy index $β=β_0^{2,0}$, points towards a significant difference between random packings of spheres and those of oblate ellipsoids. While the average cell shape $β$ of all cells with a given value of $ϕ_l$ is very similar in dense and loose jammed sphere packings, the structure of dense and loose ellipsoid packings differs substantially such that this does not hold true. This non-universality has implications for our understanding of jamming of aspherical particles.

cond-mat.soft↗

arXiv2015

Unimodular lattice triangulations as small-world and scale-free random graphs

Real-world networks, e.g. the social relations or world-wide-web graphs, exhibit both small-world and scale-free behaviour. We interpret lattice triangulations as planar graphs by identifying triangulation vertices with graph nodes and one-dimensional simplices with edges. Since these triangulations are ergodic with respect to a certain Pachner flip, applying different Monte-Carlo simulations enables us to calculate average properties of random triangulations, as well as canonical ensemble averages using an energy functional that is approximately the variance of the degree distribution. All considered triangulations have clustering coefficients comparable with real world graphs, for the canonical ensemble there are inverse temperatures with small shortest path length independent of system size. Tuning the inverse temperature to a quasi-critical value leads to an indication of scale-free behaviour for degrees $k \geq 5$. Using triangulations as a random graph model can improve the understanding of real-world networks, especially if the actual distance of the embedded nodes becomes important.

cond-mat.dis-nn↗

arXiv2014

Entropy of unimodular Lattice Triangulations

Triangulations are important objects of study in combinatorics, finite element simulations and quantum gravity, where its entropy is crucial for many physical properties. Due to their inherent complex topological structure even the number of possible triangulations is unknown for large systems. We present a novel algorithm for an approximate enumeration which is based on calculations of the density of states using the Wang-Landau flat histogram sampling. For triangulations on two-dimensional integer lattices we achive excellent agreement with known exact numbers of small triangulations as well as an improvement of analytical calculated asymptotics. The entropy density is $C=2.196(3)$ consistent with rigorous upper and lower bounds. The presented numerical scheme can easily be applied to other counting and optimization problems.

physics.comp-ph↗

arXiv2014

Deriving fundamental measure theory from the virial series: Consistency with the zero-dimensional limit

Fundamental measure theory (FMT) for hard particles has great potential for predicting the phase behavior of colloidal and nanometric shapes. The modern versions of FMT are usually derived from the zero-dimensional limit, a system of at most one particle confined in a collection of cavities in the limit that all cavities shrink to the size of the particle. In [Phys. Rev. E 85, 041150 (2012)], a derivation from an approximated and resummed virial expansion was presented, whose result was not fully consistent with the FMT from the zero-dimensional limit. Here we improve upon this derivation and obtaining exactly the same FMT functional as was obtained earlier from the zero-dimensional limit. As a result, further improvements of FMT based on the virial expansion can now be formulated, some of which we suggest in the outlook.

cond-mat.soft↗

arXiv2014

Simulation of a Hard-Spherocylinder Liquid Crystal with the pe

The pe physics engine is validated through the simulation of a liquid crystal model system consisting of hard spherocylinders. For this purpose we evaluate several characteristic parameters of this system, namely the nematic order parameter, the pressure, and the Frank elastic constants. We compare these to the values reported in literature and find a very good agreement, which demonstrates that the pe physics engine can accurately treat such densely packed particle systems. Simultaneously we are able to examine the influence of finite size effects, especially on the evaluation of the Frank elastic constants, as we are far less restricted in system size than earlier simulations.

physics.comp-ph↗

arXiv2014

Dynamical States in Driven Colloidal Liquid Crystals

We study a model colloidal liquid crystal consisting of hard spherocylinders under the influence of an external aligning potential by Langevin dynamics simulation. The external field that rotates in a plane acts on the orientation of the individual particles and induces a variety of collective nonequilibrium states. We characterize these states by the time-resolved orientational distribution of the particles and explain their origin using the single particle behavior. By varying the external driving frequency and the packing fraction of the spherocylinders we construct the dynamical state diagram.

cond-mat.soft↗

arXiv2014

Surface tension of isotropic-nematic interfaces: Fundamental Measure Theory for hard spherocylinders

A fluid constituted of hard spherocylinders is studied using a density functional theory for non-spherical hard particles, which can be written as a function of weighted densities. This is based on an extended deconvolution of the Mayer $f$-function for arbitrarily shaped convex hard bodies in tensorial weight functions, which depend each only on the shape and orientation of a single particle. In the course of an examination of the isotropic- nematic interface at coexistence the functional is applied to anisotropic and inhomogeneous problems for the first time. We find good qualitative agreement with other theoretical predictions and also with Monte-Carlo simulations.

cond-mat.soft↗

arXiv2013

Percolation transition by random vertex splitting of diamond networks?

We propose a statistical model defined on the three-dimensional diamond network where the splitting of randomly selected nodes leads to a spatially disordered network, with decreasing degree of connectivity. The terminal state, that is reached when all nodes have been split, is a dense configuration of self-avoiding walks on the diamond network. Starting from the crystallographic diamond network, each of the four-coordinated nodes is replaced with probability p by a pair of two edges, each connecting a pair of the adjacent vertices. For all values of 0 <= p <= 1 the network percolates, yet the fraction fp of the system that belongs to a percolating cluster drops sharply at pc=1 to a finite value. This transition has the signature of a phase transition with scaling exponents for p close to pc that are different from the critical exponents of the second-order phase transition of standard percolation models. As is the case for percolation transitions, this transition significantly affects the mechanical properties of linear-elastic realisations, obtained by replacing edges with solid circular struts to give an effective density phi. Finite element methods demonstrate that, as a low-density cellular structure, the bulk modulus K shows a cross-over from a compression-dominated behaviour (with K proportional to phi) at p=0 to a bending-dominated behaviour (with quadratic dependence of K on phi) at p=1.

cond-mat.dis-nn↗