arXiv subjects

Michael A. Klatt

Publications and source records attributed to Michael A. Klatt.

At least 19 recordsLinked to original sources

arXiv2026

Persistence of asymptotic variance under transport: from hyperfluctuation to stealthy hyperuniformity

We introduce $p$-uniformity to characterize the scaling of density fluctuations in spatial random systems in $\mathbb R^d$, ranging from hyperfluctuation to stealthy hyperuniformity. Our central theorem establishes sufficient conditions to preserve $p$-uniformity under transport. The first condition, a finite $(d+p)$-th moment of the transport distance, allows for a Taylor expansion of the transport. The second condition controls the corresponding terms. We thus solve a previously stated open problem; indeed we extend it, since our result applies to a general $p$-uniform source in any dimension, and the source and transport may be dependent. As an application, we construct new classes of point processes that are isotropic and $p$-uniform with arbitrarily high $p$, and that can be simulated in linear time. We thus achieve three-dimensional isotropic hyperuniform samples with an unprecedented system size of $10^9$ points. We conclude with an outlook on a converse statement.

math.PR↗

arXiv2026

Towards stealthy hyperuniform networks with optimal isotropic complete photonic band gaps using a novel inverse design procedure

We present a two-stage inverse design procedure for producing disordered stealthy hyperuniform trivalent photonic networks in two dimensions with isotropic complete photonic band gaps (PBGs) blocking light regardless of direction or polarization (TE or TM) over a wide frequency range. Most ordinary disordered systems fail to maintain complete PBGs as system size increases. The only known exceptions that remain open in the largest simulations have been generated by mapping stealthy hyperuniform point patterns into trivalent networks. However, the resulting networks are not truly stealthy hyperuniform two-phase media. Although their PBGs remain open, they are relatively narrow due to limited overlap between the TE and TM band gaps and broad band tails caused by localized defect states. By contrast, our two-stage inverse design aims to make the final network itself stealthy hyperuniform, achieving unprecedented near-optimal overlap between the TE and TM band gaps and a small defect state density at the band edges. We obtain not only single realizations with large PBGs, but a striking homogeneity across a large ensemble, effectively probing a network with 100,000 vertices. This ensemble-based band gap is comparable in width to the complete PBG of an anisotropic honeycomb photonic crystal with the same network parameters and nearly an order of magnitude wider than the previously widest known isotropic complete PBGs. Our designs can be fabricated using additive manufacturing, offering new pathways to manipulate electromagnetic waves for photonic technologies.

physics.optics↗

arXiv2026

Minkowski tensors for point clouds and voxelized data: robust, asymptotically unbiased estimators

Minkowski tensors, also known as tensor valuations, provide robust $n$-point information for a wide range of random spatial structures. Local estimators for point clouds, e.g., representing voxelized data, however, are unavoidably biased even in the limit of infinitely high resolution. Here, we substantially improve a recently proposed, asymptotically unbiased algorithm to estimate Minkowski tensors from point clouds. Our improved algorithm is more robust and efficient. Moreover we generalize the theoretical foundations for an asymptotically bias-free estimation of the interfacial tensors, among others, to the case of finite unions of compact sets with positive reach, which is relevant for many applications like rough surfaces or composite materials. As a realistic test case of random spatial structures, we consider random (beta) polytopes. We first derive explicit expressions of the expected Minkowski tensors, which we then compare to our simulation results. We obtain precise estimates with relative errors of a few percent for practically relevant resolutions. Finally, we apply our methods to real data of metallic grains and nanorough surfaces, and we provide an open-source python package, which works in any dimension.

math.ST↗

arXiv2026

A genuine test for hyperuniformity

We introduce a rigorous and sensitive significance test for hyperuniformity that yields reliable results even from a single sample. Our approach is based on a detailed analysis of the empirical Fourier transform of a stationary point process in $\mathbb{R}^d$. For large system sizes, we derive the asymptotic covariances and establish a multivariate central limit theorem (CLT) for these empirical Fourier transforms. Their absolute square value, the scattering intensity, is then used as the standard estimator of the structure factor. The above CLT holds for a preferably large class of point processes, and whenever this is the case, the scattering intensity satisfies a multivariate limit theorem as well. Hence, we can use the likelihood ratio principle to test for hyperuniformity. Remarkably, the asymptotic distribution of the resulting test statistic is universal under the null hypothesis of hyperuniformity. We obtain its explicit form from simulations with very high accuracy. The novel test precisely keeps a nominal significance level for hyperuniform models, and it rejects non-hyperuniform examples with high power even in borderline cases. Moreover, it does so given only a single sample with a practically relevant system size.

math.ST↗

arXiv2026

Hyperuniformity of Weighted Particle Systems

Hyperuniform particle arrangements are characterized by a local number variance that grows more slowly than the volume of the observation window. We generalize this concept to describe particle systems in which particles carry weights: internal degrees of freedom such as scalars, vectors, pseudovectors, directors, tensors, or extrinsic local attributes. Our generalization extends hyperuniformity from fluctuations in particle positions to fluctuations in the spatial distribution of weights. We derive generalized weighted pair correlation, autocovariance, and spectral functions, and show their relation to the local variance in weighted many-particle systems. Applying this formalism to bond-orientational ordered phases, dipolar liquid water, Voronoi-cell volumes, and certain ionic liquids, we demonstrate that hyperuniformity in the particle system does not necessarily translate to hyperuniformity of the weighted system. In fact, cases exist where a hyperuniform particle system becomes antihyperuniform when weighted, and others where nonhyperuniform or antihyperuniform particle systems yield hyperuniform weighted systems. This theoretical framework provides a road map for quantifying large-scale fluctuations in weighted many-particle systems, offering a powerful tool for identifying systems with novel physical properties.

cond-mat.stat-mech↗

arXiv2025

Transparency versus Anderson localization in one-dimensional disordered stealthy hyperuniform layered media

We present numerical simulations of disordered stealthy hyperuniform layered media ranging up to 10,000 thin slabs of high-dielectric constant separated by intervals of low dielectric constant that show no apparent evidence of Anderson localization of electromagnetic waves or deviations from transparency for a continuous band of frequencies ranging from zero up to some value $ω_T$. The results are consistent with the strong-contrast formula including its tight upper bound on $ω_T$ and with previous simulations on much smaller systems. We utilize a transfer matrix method to compute the Lyaponov exponents, which we show is a more reliable method for detecting Anderson localization by applying it to a range of systems with common types of disorder known to exhibit localization, such as perturbed periodic lattices. The Lyaponov exponents for these systems with ordinary disorder show clear evidence of localization, in contrast to the cases of perfectly periodically spaced slabs and disordered stealthy hyperuniform layered systems. As with any numerical study, one should be cautious about drawing definitive conclusions. There remains the challenge of determining whether one-dimensional disordered stealthy hyperuniform layered media possess a finite localization length on some scale much larger than our already large system size or, alternatively, are exceptions to the standard Anderson localization theorems.

cond-mat.dis-nn↗

arXiv2025

Invariant transports of stationary random measures: asymptotic variance, hyperuniformity, and examples

We consider invariant transports of stationary random measures on $\mathbb{R}^d$ and establish natural mixing criteria that guarantee persistence of asymptotic variances. To check our mixing assumptions, which are based on two-point Palm probabilities, we combine factorial moment expansion with stopping set techniques, among others. We complement our results by providing formulas for the Bartlett spectral measure of the destinations. We pay special attention to the case of a vanishing asymptotic variance, known as hyperuniformity. By constructing suitable transports from a hyperuniform source we are able to rigorously establish hyperuniformity for many point processes and random measures. On the other hand, our method can also refute hyperuniformity. For instance, we show that finitely many steps of Lloyd's algorithm or of a random organization model preserve the asymptotic variance if we start from a Poisson process or a point process with exponentially fast decaying correlation. Finally, we define a hyperuniformerer that turns any ergodic point process with finite intensity into a hyperuniform process by randomizing each point within its cell of a fair partition.

math.PR↗

arXiv2024

Impact of geometry on chemical analysis exemplified for photoelectron spectroscopy of black silicon

For smooth surfaces, chemical composition can be readily analyzed using various spectroscopic techniques, a prominent example is X-ray photoelectron spectroscopy (XPS), where the relative proportions of the elements are mainly determined by the intensity ratio of the element-specific photoelectrons. However, this analysis becomes more complex for nanorough surfaces like black silicon (b-Si) due to the geometry's steep slopes, which mimic local variations in emission angles. In this study, we explicitly quantify this effect through an integral geometric analysis using Minkowski tensors, correlating XPS chemical data with topographical information from Atomic Force Microscopy (AFM). This approach yields reliable estimates of layer thicknesses for nanorough surfaces. For b-Si, we found that the oxide layer is approximately 50% thicker than the native oxide layer on a standard Si wafer. This study underscores the significant impact of nanoscale geometries on chemical property analysis.

cond-mat.mes-hall↗

arXiv2024

Hole-Based Stealthy Hyperuniform Semiconductor Metamaterials for the Mid-Infrared

Stealthy hyperuniform heterostructures are a novel type of metamaterial with the potential for optical image processing at angles away from normal incidence. These metamaterials show analogous properties to photonic crystals while circumventing the spatial anisotropy often hindering the latter's use. In this paper, we have successfully designed, fabricated, and characterized a hole-based stealthy hyperuniform structure on a quantum cascade layer substrate. The infrared spectral data reveal a sizable gap-midgap ratio of 9.8% for a photonic band gap around $12.0 ~μm$ in the form of an enhanced reflection region for increasing incidence angles. The stealthy hyperuniform metamaterial also showed spatial isotropy by its unchanging reflection spectrum for all in-plane rotational angle measurements.

physics.optics↗

arXiv2024

Local number fluctuations in ordered and disordered phases of water across temperatures: Higher-order moments and degrees of tetrahedrality

The isothermal compressibility (i.e., the asymptotic number variance) of equilibrium liquid water as a function of temperature is minimal near ambient conditions. This anomalous non-monotonic temperature dependence is due to a balance between thermal fluctuations and the formation of tetrahedral hydrogen-bond networks. Since tetrahedrality is a many-body property, it will also influence the higher-order moments of density fluctuations, including the skewness and kurtosis. To gain a more complete picture, we examine these higher-order moments that encapsulate many-body correlations using a recently developed, advanced platform for local density fluctuations. We study an extensive set of simulated phases of water across a range of temperatures (80 K to 1600 K) with various degrees of tetrahedrality, including ice phases, equilibrium liquid water, supercritical water, and disordered nonequilibrium quenches. We find clear signatures of tetrahedrality in the higher-order moments, including the skewness and excess kurtosis, that scale for all cases with the degree of tetrahedrality. More importantly, this scaling behavior leads to non-monotonic temperature dependencies in the higher-order moments for both equilibrium and non-equilibrium phases. Specifically, at near-ambient conditions, the higher-order moments vanish most rapidly for large length scales, and the distribution quickly converges to a Gaussian in our metric. However, at non-ambient conditions, higher-order moments vanish more slowly and hence become more relevant especially for improving information-theoretic approximations of hydrophobic solubility. The temperature non-monotonicity that we observe in the full distribution across length-scales could shed light on water's nested anomalies, i.e., reveal new links between structural, dynamic, and thermodynamic anomalies.

cond-mat.stat-mech↗

arXiv2024

Universal Hyperuniformity in Active Field Theories

We show that dry scalar-order active field theories (AFTs) are universally hyperuniform, i.e., density fluctuations are anomalously suppressed in the long-time limit regardless of the integrability or functional form of the active contributions up to third order in gradient terms. These AFTs include Active model B, Active model B+, and effective Cahn-Hilliard models. Moreover, density variances and spectral densities are virtually indistinguishable from that of passive phase-separated hyperuniform fields. Higher moments of the density fluctuations, however, reveal activity-dependent higher-order correlations that are not captured by conventional two-point measures that characterize hyperuniformity.

cond-mat.stat-mech↗

arXiv2024

Emergence of Disordered Hyperuniformity in Melts of Linear Diblock Copolymers

Disordered hyperuniform (DHU) systems are recently discovered exotic states of matter, where (normalized) large-scale density fluctuations are completely suppressed as in crystals, even though the systems are isotropic and lack conventional long-range order. Despite recent success, realizing such systems using bottom-up approaches remains challenging. Here, we study the large-scale behavior of neat melts of linear diblock copolymers using large-cell self-consistent field theory (SCFT) simulations. We initialize SCFT simulations using point patterns that correspond to the local energy minimum of the so-called Quantizer energy, a geometric functional related to the free energy of copolymeric self-assemblies. Upon relaxation via the SCFT simulations, we obtain a new class of metastable disordered micelle mesophases that are hyperuniform. Moreover, we find that DHU micelle mesophases possess very similar free energies to the thermodynamically stable body-centered cubic sphere mesophases and are also much more favorable energetically than previously obtained liquid-like packings. Our findings shed light on the design of novel disordered hyperuniform materials using bottom-up approaches, and suggest new possibilities for technological applications, e.g., novel non-iridescent structural colors.

cond-mat.soft↗

arXiv2023

Almost sure convergence and second moments of geometric functionals of fractal percolation

We determine almost sure limits of rescaled intrinsic volumes of the construction steps of fractal percolation in $\mathbb{R}^d$ for any dimension $d\geq 1$. We observe a factorization of these limit variables which allows, in particular, to determine their expectations and covariance structure. We also show convergence of rescaled expectations and variances of the intrinsic volumes of the construction steps to expectations and variances of the limit variables and give rates for this convergence in some cases. These results significantly extend our previous work that addressed only limits of expectations of intrinsic volumes.

math.PR↗

arXiv2022

Active particles using reinforcement learning to navigate in complex motility landscapes

As the length scales of the smallest technology continue to advance beyond the micron scale it becomes increasingly important to equip robotic components with the means for intelligent and autonomous decision making with limited information. With the help of a tabular Q-learning algorithm, we design a model for training a microswimmer, to navigate quickly through an environment given by various different scalar motility fields, while receiving a limited amount of local information. We compare the performances of the microswimmer, defined via time of first passage to a target, with performances of suitable reference cases. We show that the strategy obtained with our reinforcement learning model indeed represents an efficient navigation strategy, that outperforms the reference cases. By confronting the swimmer with a variety of unfamiliar environments after the finalised training, we show that the obtained strategy generalises to different classes of random fields.

cond-mat.soft↗

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↗

arXiv2021

Critical pore radius and transport properties of disordered hard- and overlapping-sphere models

Descriptors that characterize the geometry and topology of the pore space of porous media are intimately linked to their transport properties. We quantify such descriptors, including pore-size functions and the critical pore radius $δ_c$, for four different models: maximally random jammed sphere packings, overlapping spheres, equilibrium hard spheres, and inherent structures of the quantizer energy. For precise estimates of the percolation thresholds, we use a strict relation of the void percolation around sphere configurations to weighted bond percolation on the corresponding Voronoi networks. We use the Newman-Ziff algorithm to determine the percolation threshold using universal properties of the cluster size distribution. Often, $δ_c$ is used as the key characteristic length scale that determines the fluid permeability $k$. A recent study [Torquato. Adv. Wat. Resour. 140, 103565 (2020)] suggested for porous media with a well-connected pore space an alternative estimate of $k$ based on the second moment of the pore size $\langleδ^2\rangle$. Here, we confirm that, for all porosities and all models considered, $δ_c^2$ is to a good approximation proportional to $\langleδ^2\rangle$. However, unlike $\langleδ^2\rangle$, the permeability estimate based on $δ_c^2$ does not predict the correct ranking of $k$ for our models. Thus, we confirm $\langleδ^2\rangle$ to be a promising candidate for convenient and reliable estimates of $k$ for porous media with a well-connected pore space. Moreover, we compare the fluid permeability of our models with varying degrees of order, as measured by the $τ$ order metric. We find that (effectively) hyperuniform models tend to have lower values of $k$ than their nonhyperuniform counterparts. Our findings could facilitate the design of porous media with desirable transport properties via targeted pore statistics.

cond-mat.soft↗

arXiv2021

Gap Sensitivity Reveals Universal Behaviors in Optimized Photonic Crystal and Disordered Networks

Through an extensive series of high-precision numerical computations of the optimal complete photonic band gap (PBG) as a function of dielectric contrast $α$ for a variety of crystal and disordered heterostructures, we reveal striking universal behaviors of the gap sensitivity $\mathcal{S}(α)\equiv dΔ(α)/dα$, the first derivative of the optimal gap-to-midgap ratio $Δ(α)$. In particular, for all our crystal networks, $\mathcal{S}(α)$ takes a universal form that is well approximated by the analytic formula for a one-dimensional quarter-wave stack, $\mathcal{S}_{\text{QWS}}(α)$. Even more surprisingly, the values of $\mathcal{S}(α)$ for our disordered networks converge to $\mathcal{S}_{\text{QWS}}(α)$ for sufficiently large $α$. A deeper understanding of the simplicity of this universal behavior may provide fundamental insights about PBG formation and guidance in the design of novel photonic heterostructures.

physics.optics↗

arXiv2020

Low-temperature statistical mechanics of the QuanTizer problem: fast quenching and equilibrium cooling of the three-dimensional Voronoi Liquid

The Quantizer problem is a tessellation optimisation problem where point configurations are identified such that the Voronoi cells minimise the second moment of the volume distribution. While the ground state (optimal state) in 3D is almost certainly the body-centered cubic lattice, disordered and effectively hyperuniform states with energies very close to the ground state exist that result as stable states in an evolution through the geometric Lloyd's algorithm [Klatt et al. Nat. Commun., 10, 811 (2019)]. When considered as a statistical mechanics problem at finite temperature, the same system has been termed the 'Voronoi Liquid' by [Ruscher et al. EPL 112, 66003 (2015)]. Here we investigate the cooling behaviour of the Voronoi liquid with a particular view to the stability of the effectively hyperuniform disordered state. As a confirmation of the results by Ruscher et al., we observe, by both molecular dynamics and Monte Carlo simulations, that upon slow quasi-static equilibrium cooling, the Voronoi liquid crystallises from a disordered configuration into the body-centered cubic configuration. By contrast, upon sufficiently fast non-equilibrium cooling (and not just in the limit of a maximally fast quench) the Voronoi liquid adopts similar states as the effectively hyperuniform inherent structures identified by Klatt et al. and prevents the ordering transition into a BCC ordered structure. This result is in line with the geometric intuition that the geometric Lloyd's algorithm corresponds to a type of fast quench.

cond-mat.soft↗