151 research outputs found

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

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

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

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

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

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

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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