268 research outputs found
Hyperuniformity and its Generalizations
Disordered many-particle hyperuniform systems are exotic amorphous states
characterized by anomalous suppression of large-scale density fluctuations.
Here we substantially broaden the hyperuniformity concept along four different
directions. This includes generalizations to treat fluctuations in the
interfacial area in heterogeneous media and surface-area driven evolving
microstructures, random scalar fields, divergence-free random vector fields, as
well as statistically anisotropic many-particle systems and two-phase media.
Interfacial-area fluctuations play a major role in characterizing the
microstructure of two-phase systems , physical properties that intimately
depend on the geometry of the interface, and evolving two-phase microstructures
that depend on interfacial energies (e.g., spinodal decomposition). In the
instances of divergence-free random vector fields and statistically anisotropic
structures, we show that the standard definition of hyperuniformity must be
generalized such that it accounts for the dependence of the relevant spectral
functions on the direction in which the origin in Fourier space
(nonanalyticities at the origin). Using this analysis, we place some well-known
energy spectra from the theory of isotropic turbulence in the context of this
generalization of hyperuniformity. We show that there exist many-particle
ground-state configurations in which directional hyperuniformity imparts exotic
anisotropic physical properties (e.g., elastic, optical and acoustic
characteristics) to these states of matter. Such tunablity could have
technological relevance for manipulating light and sound waves in ways
heretofore not thought possible. We show that disordered many-particle systems
that respond to external fields (e.g., magnetic and electric fields) are a
natural class of materials to look for directional hyperuniformity.Comment: In pres
Hyperuniformity of generalized random organization models
Studies of random organization models of monodisperse spherical particles
have shown that a hyperuniform state is achievable when the system goes through
an absorbing phase transition to a critical state. Here we investigate to what
extent hyperuniformity is preserved when the model is generalized to particles
with a size distribution and/or nonspherical shapes. We begin by examining
binary disks in two dimensions and demonstrate that their critical states are
hyperuniform as two-phase media, but not hyperuniform nor multihyperuniform as
point patterns formed by the particle centroids. We further confirm the
generality of our findings by studying particles with a continuous size
distribution. Finally, to study the effect of rotational degrees of freedom, we
extend our model to noncircular particles, namely, hard rectangles with various
aspect ratios, including the hard-needle limit. Although these systems exhibit
only short-range orientational order, hyperuniformity is still preserved. Our
analysis reveals that the redistribution of the "mass" of the particles rather
than the particle centroids is central to this dynamical process. The
consideration of the "active volume fraction" of generalized random
organization models may help to resolve which universality class they belong to
and hence may lead to a deeper theoretical understanding of absorbing-state
models. Our results suggest that general particle systems subject to random
organization can be a robust way to fabricate a wide class of hyperuniform
states of matter by tuning the structures via different particle-size and
-shape distributions. This in turn potentially enables the creation of
multifunctional hyperuniform materials with desirable optical, transport, and
mechanical properties
Random Scalar Fields and Hyperuniformity
Disordered many-particle hyperuniform systems are exotic amorphous states of
matter that lie between crystals and liquids. Hyperuniform systems have
attracted recent attention because they are endowed with novel transport and
optical properties. Recently, the hyperuniformity concept has been generalized
to characterize scalar fields, two-phase media and random vector fields. In
this paper, we devise methods to explicitly construct hyperuniform scalar
fields. We investigate explicitly spatial patterns generated from Gaussian
random fields, which have been used to model the microwave background radiation
and heterogeneous materials, the Cahn-Hilliard equation for spinodal
decomposition, and Swift-Hohenberg equations that have been used to model
emergent pattern formation, including Rayleigh-B{\' e}nard convection. We show
that the Gaussian random scalar fields can be constructed to be hyperuniform.
We also numerically study the time evolution of spinodal decomposition patterns
and demonstrate that these patterns are hyperuniform in the scaling regime.
Moreover, we find that labyrinth-like patterns generated by the Swift-Hohenberg
equation are effectively hyperuniform. We show that thresholding a hyperuniform
Gaussian random field to produce a two-phase random medium tends to destroy the
hyperuniformity of the progenitor scalar field. We then propose guidelines to
achieve effectively hyperuniform two-phase media derived from thresholded
non-Gaussian fields. Our investigation paves the way for new research
directions to characterize the large-structure spatial patterns that arise in
physics, chemistry, biology and ecology. Moreover, our theoretical results are
expected to guide experimentalists to synthesize new classes of hyperuniform
materials with novel physical properties via coarsening processes and using
state-of-the-art techniques, such as stereolithography and 3D printing.Comment: 16 pages, 18 figure
Precise algorithms to compute surface correlation functions of two-phase heterogeneous media and their applications
The quantitative characterization of the microstructure of random
heterogeneous media in -dimensional Euclidean space via a
variety of -point correlation functions is of great importance, since the
respective infinite set determines the effective physical properties of the
media. In particular, surface-surface and surface-void
correlation functions (obtainable from radiation scattering experiments)
contain crucial interfacial information that enables one to estimate transport
properties of the media (e.g., the mean survival time and fluid permeability)
and complements the information content of the conventional two-point
correlation function. However, the current technical difficulty involved in
sampling surface correlation functions has been a stumbling block in their
widespread use. We first present a concise derivation of the small-
behaviors of these functions, which are linked to the \textit{mean curvature}
of the system. Then we demonstrate that one can reduce the computational
complexity of the problem by extracting the necessary interfacial information
from a cut of the system with an infinitely long line. Accordingly, we devise
algorithms based on this idea and test them for two-phase media in continuous
and discrete spaces. Specifically for the exact benchmark model of overlapping
spheres, we find excellent agreement between numerical and exact results. We
compute surface correlation functions and corresponding local surface-area
variances for a variety of other model microstructures, including hard spheres
in equilibrium, decorated "stealthy" patterns, as well as snapshots of evolving
pattern formation processes (e.g., spinodal decomposition). It is demonstrated
that the precise determination of surface correlation functions provides a
powerful means to characterize a wide class of complex multiphase
microstructures
Effect of Dimensionality on the Continuum Percolation of Overlapping Hyperspheres and Hypercubes: II. Simulation Results and Analyses
In the first paper of this series [S. Torquato, J. Chem. Phys. {\bf 136},
054106 (2012)], analytical results concerning the continuum percolation of
overlapping hyperparticles in -dimensional Euclidean space
were obtained, including lower bounds on the percolation threshold. In the
present investigation, we provide additional analytical results for certain
cluster statistics, such as the concentration of -mers and related
quantities, and obtain an upper bound on the percolation threshold . We
utilize the tightest lower bound obtained in the first paper to formulate an
efficient simulation method, called the {\it rescaled-particle} algorithm, to
estimate continuum percolation properties across many space dimensions with
heretofore unattained accuracy. This simulation procedure is applied to compute
the threshold and associated mean number of overlaps per particle
for both overlapping hyperspheres and oriented hypercubes for . These simulations results are compared to corresponding upper
and lower bounds on these percolation properties. We find that the bounds
converge to one another as the space dimension increases, but the lower bound
provides an excellent estimate of and , even for
relatively low dimensions. We confirm a prediction of the first paper in this
series that low-dimensional percolation properties encode high-dimensional
information. We also show that the concentration of monomers dominate over
concentration values for higher-order clusters (dimers, trimers, etc.) as the
space dimension becomes large. Finally, we provide accurate analytical
estimates of the pair connectedness function and blocking function at their
contact values for any as a function of density.Comment: 24 pages, 10 figure
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