151 research outputs found

    Characterization of Maximally Random Jammed Sphere Packings: II. Correlation Functions and Density Fluctuations

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    In the first paper of this series, we introduced Voronoi correlation functions to characterize the structure of maximally random jammed (MRJ) sphere packings across length scales. In the present paper, we determine a variety of correlation functions that can be rigorously related to effective physical properties of MRJ sphere packings and compare them to the corresponding statistical descriptors for overlapping spheres and equilibrium hard-sphere systems. Such structural descriptors arise in rigorous bounds and formulas for effective transport properties, diffusion and reactions constants, elastic moduli, and electromagnetic characteristics. First, we calculate the two-point, surface-void, and surface-surface correlation functions, for which we derive explicit analytical formulas for finite hard-sphere packings. We show analytically how the contacts between spheres in the MRJ packings translate into distinct functional behaviors of these two-point correlation functions that do not arise in the other two models examined here. Then, we show how the spectral density distinguishes the MRJ packings from the other disordered systems in that the spectral density vanishes in the limit of infinite wavelengths. These packings are hyperuniform, which means that density fluctuations on large length scales are anomalously suppressed. Moreover, we study and compute exclusion probabilities and pore size distributions as well as local density fluctuations. We conjecture that for general disordered hard-sphere packings, a central limit theorem holds for the number of points within an spherical observation window. Our analysis links problems of interest in material science, chemistry, physics, and mathematics. In the third paper, we will evaluate bounds and estimates of a host of different physical properties of the MRJ sphere packings based on the structural characteristics analyzed in this paper.Comment: 25 pages, 13 Figures; corrected typos, updated reference

    Characterization of Maximally Random Jammed Sphere Packings. III. Transport and Electromagnetic Properties via Correlation Functions

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    In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here, we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical approximation formulas. These property formulas include interfacial bounds as well as universal scaling laws for the mean survival time and the fluid permeability. We also estimate the principal relaxation time associated with Brownian motion among perfectly absorbing traps. For the propagation of electromagnetic waves in the long-wavelength limit, we show that a dispersion of dielectric MRJ spheres within a matrix of another dielectric material forms, to a very good approximation, a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio. We compare the effective properties of the MRJ sphere packings to those of overlapping spheres, equilibrium hard-sphere packings, and lattices of hard spheres. Moreover, we generalize results to micro- and macroscopically anisotropic packings of spheroids with tensorial effective properties. The analytic bounds predict the qualitative trend in the physical properties associated with these structures, which provides guidance to more time-consuming simulations and experiments. They especially provide impetus for experiments to design materials with unique bulk properties resulting from hyperuniformity, including structural-color and color-sensing applications.Comment: 19 pages, 16 Figure

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

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    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

    Active particles using reinforcement learning to navigate in complex motility landscapes

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    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

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

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    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.Comment: 11 pages, 6 figure

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

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    For a smooth surface, the chemical composition can be readily evaluated by a variety of spectroscopy 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. This deduction, however, is more intricate for a nanorough surface, such as black silicon, since the steep slopes of the geometry mimic local variations of the local emission angle. Here, we explicitly quantify this effect via an integral geometric analysis, by using so-called Minkowski tensors. Thus, we match the chemical information from XPS with topographical information from atomic force microscopy (AFM). Our method provides reliable estimates of layer thicknesses for nanorough surfaces. For our black silicon samples, we found that the oxide layer thickness is on average comparable to that of a native oxide layer. Our study highlights the impact of complex geometries at the nanoscale on the analysis of chemical properties with implications for a broad class of spectroscopy techniques
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