551 research outputs found
Statistical properties of determinantal point processes in high-dimensional Euclidean spaces
The goal of this paper is to quantitatively describe some statistical
properties of higher-dimensional determinantal point processes with a primary
focus on the nearest-neighbor distribution functions. Toward this end, we
express these functions as determinants of matrices and then
extrapolate to . This formulation allows for a quick and accurate
numerical evaluation of these quantities for point processes in Euclidean
spaces of dimension . We also implement an algorithm due to Hough \emph{et.
al.} \cite{hough2006dpa} for generating configurations of determinantal point
processes in arbitrary Euclidean spaces, and we utilize this algorithm in
conjunction with the aforementioned numerical results to characterize the
statistical properties of what we call the Fermi-sphere point process for to 4. This homogeneous, isotropic determinantal point process, discussed
also in a companion paper \cite{ToScZa08}, is the high-dimensional
generalization of the distribution of eigenvalues on the unit circle of a
random matrix from the circular unitary ensemble (CUE). In addition to the
nearest-neighbor probability distribution, we are able to calculate Voronoi
cells and nearest-neighbor extrema statistics for the Fermi-sphere point
process and discuss these as the dimension is varied. The results in this
paper accompany and complement analytical properties of higher-dimensional
determinantal point processes developed in \cite{ToScZa08}.Comment: 42 pages, 17 figure
Modeling Heterogeneous Materials via Two-Point Correlation Functions: II. Algorithmic Details and Applications
In the first part of this series of two papers, we proposed a theoretical
formalism that enables one to model and categorize heterogeneous materials
(media) via two-point correlation functions S2 and introduced an efficient
heterogeneous-medium (re)construction algorithm called the "lattice-point"
algorithm. Here we discuss the algorithmic details of the lattice-point
procedure and an algorithm modification using surface optimization to further
speed up the (re)construction process. The importance of the error tolerance,
which indicates to what accuracy the media are (re)constructed, is also
emphasized and discussed. We apply the algorithm to generate three-dimensional
digitized realizations of a Fontainebleau sandstone and a boron
carbide/aluminum composite from the two- dimensional tomographic images of
their slices through the materials. To ascertain whether the information
contained in S2 is sufficient to capture the salient structural features, we
compute the two-point cluster functions of the media, which are superior
signatures of the micro-structure because they incorporate the connectedness
information. We also study the reconstruction of a binary laser-speckle pattern
in two dimensions, in which the algorithm fails to reproduce the pattern
accurately. We conclude that in general reconstructions using S2 only work well
for heterogeneous materials with single-scale structures. However, two-point
information via S2 is not sufficient to accurately model multi-scale media.
Moreover, we construct realizations of hypothetical materials with desired
structural characteristics obtained by manipulating their two-point correlation
functions.Comment: 35 pages, 19 figure
Densest local packing diversity. II. Application to three dimensions
The densest local packings of N three-dimensional identical nonoverlapping
spheres within a radius Rmin(N) of a fixed central sphere of the same size are
obtained for selected values of N up to N = 1054. In the predecessor to this
paper [A.B. Hopkins, F.H. Stillinger and S. Torquato, Phys. Rev. E 81 041305
(2010)], we described our method for finding the putative densest packings of N
spheres in d-dimensional Euclidean space Rd and presented those packings in R2
for values of N up to N = 348. We analyze the properties and characteristics of
the densest local packings in R3 and employ knowledge of the Rmin(N), using
methods applicable in any d, to construct both a realizability condition for
pair correlation functions of sphere packings and an upper bound on the maximal
density of infinite sphere packings. In R3, we find wide variability in the
densest local packings, including a multitude of packing symmetries such as
perfect tetrahedral and imperfect icosahedral symmetry. We compare the densest
local packings of N spheres near a central sphere to minimal-energy
configurations of N+1 points interacting with short-range repulsive and
long-range attractive pair potentials, e.g., 12-6 Lennard-Jones, and find that
they are in general completely different, a result that has possible
implications for nucleation theory. We also compare the densest local packings
to finite subsets of stacking variants of the densest infinite packings in R3
(the Barlow packings) and find that the densest local packings are almost
always most similar, as measured by a similarity metric, to the subsets of
Barlow packings with the smallest number of coordination shells measured about
a single central sphere, e.g., a subset of the FCC Barlow packing. We
additionally observe that the densest local packings are dominated by the
spheres arranged with centers at precisely distance Rmin(N) from the fixed
sphere's center.Comment: 45 pages, 18 figures, 2 table
Stochastic reconstruction of sandstones
A simulated annealing algorithm is employed to generate a stochastic model
for a Berea and a Fontainebleau sandstone with prescribed two-point probability
function, lineal path function, and ``pore size'' distribution function,
respectively. We find that the temperature decrease of the annealing has to be
rather quick to yield isotropic and percolating configurations. A comparison of
simple morphological quantities indicates good agreement between the
reconstructions and the original sandstones. Also, the mean survival time of a
random walker in the pore space is reproduced with good accuracy. However, a
more detailed investigation by means of local porosity theory shows that there
may be significant differences of the geometrical connectivity between the
reconstructed and the experimental samples.Comment: 12 pages, 5 figure
Morphology and thermal conductivity of model organic aerogels
The intersection volume of two independent 2-level cut Gaussian random fields
is proposed to model the open-cell microstructure of organic aerogels. The
experimentally measured X-ray scattering intensity, surface area and solid
thermal conductivity of both polymeric and colloidal organic aerogels can be
accounted for by the model.Comment: 5 pages. RevTex with 4 encapsulated figures. Higher resolution
figures have been submitted for publication. To be published in Phys. Rev. E
(Rapid Comm.). email, [email protected]
Robust Algorithm to Generate a Diverse Class of Dense Disordered and Ordered Sphere Packings via Linear Programming
We have formulated the problem of generating periodic dense paritcle packings
as an optimization problem called the Adaptive Shrinking Cell (ASC) formulation
[S. Torquato and Y. Jiao, Phys. Rev. E {\bf 80}, 041104 (2009)]. Because the
objective function and impenetrability constraints can be exactly linearized
for sphere packings with a size distribution in -dimensional Euclidean space
, it is most suitable and natural to solve the corresponding ASC
optimization problem using sequential linear programming (SLP) techniques. We
implement an SLP solution to produce robustly a wide spectrum of jammed sphere
packings in for and with a diversity of disorder
and densities up to the maximally densities. This deterministic algorithm can
produce a broad range of inherent structures besides the usual disordered ones
with very small computational cost by tuning the radius of the {\it influence
sphere}. In three dimensions, we show that it can produce with high probability
a variety of strictly jammed packings with a packing density anywhere in the
wide range . We also apply the algorithm to generate various
disordered packings as well as the maximally dense packings for
and 6. Compared to the LS procedure, our SLP protocol is able to ensure that
the final packings are truly jammed, produces disordered jammed packings with
anomalously low densities, and is appreciably more robust and computationally
faster at generating maximally dense packings, especially as the space
dimension increases.Comment: 34 pages, 6 figure
Designed Interaction Potentials via Inverse Methods for Self-Assembly
We formulate statistical-mechanical inverse methods in order to determine
optimized interparticle interactions that spontaneously produce target
many-particle configurations. Motivated by advances that give experimentalists
greater and greater control over colloidal interaction potentials, we propose
and discuss two computational algorithms that search for optimal potentials for
self-assembly of a given target configuration. The first optimizes the
potential near the ground state and the second near the melting point. We begin
by applying these techniques to assembling open structures in two dimensions
(square and honeycomb lattices) using only circularly symmetric pair
interaction potentials ; we demonstrate that the algorithms do indeed cause
self-assembly of the target lattice. Our approach is distinguished from
previous work in that we consider (i) lattice sums, (ii) mechanical stability
(phonon spectra), and (iii) annealed Monte Carlo simulations. We also devise
circularly symmetric potentials that yield chain-like structures as well as
systems of clusters.Comment: 28 pages, 23 figure
Hyperuniformity on spherical surfaces
In this work we present a study on the characterization of ordered and
disordered hyperuniform point distributions on spherical surfaces. In spite of
the extensive literature on disordered hyperuniform systems in Euclidean
geometries, to date few works have dealt with the problem of hyperuniformity in
curved spaces. As a matter of fact, some systems that display disordered
hyperuniformity, like the space distribution of photoreceptors in avian retina,
actually occur on curved surfaces. Here we will focus on the local particle
number variance and its dependence on the size of the sampling window (which we
take to be a spherical cap) for regular and uniform point distributions, as
well as for equilibrium configurations of fluid particles interacting through
Lennard-Jones, dipole-dipole and charge-charge potentials. We will show how the
scaling of the local number variance enables the characterization of
hyperuniform point patterns also on spherical surfaces
Density of States for a Specified Correlation Function and the Energy Landscape
The degeneracy of two-phase disordered microstructures consistent with a
specified correlation function is analyzed by mapping it to a ground-state
degeneracy. We determine for the first time the associated density of states
via a Monte Carlo algorithm. Our results are described in terms of the
roughness of the energy landscape, defined on a hypercubic configuration space.
The use of a Hamming distance in this space enables us to define a roughness
metric, which is calculated from the correlation function alone and related
quantitatively to the structural degeneracy. This relation is validated for a
wide variety of disordered systems.Comment: Accepted for publication in Physical Review Letter
Toward the Jamming Threshold of Sphere Packings: Tunneled Crystals
We have discovered a new family of three-dimensional crystal sphere packings
that are strictly jammed (i.e., mechanically stable) and yet possess an
anomalously low density. This family constitutes an uncountably infinite number
of crystal packings that are subpackings of the densest crystal packings and
are characterized by a high concentration of self-avoiding "tunnels" (chains of
vacancies) that permeate the structures. The fundamental geometric
characteristics of these tunneled crystals command interest in their own right
and are described here in some detail. These include the lattice vectors (that
specify the packing configurations), coordination structure, Voronoi cells, and
density fluctuations. The tunneled crystals are not only candidate structures
for achieving the jamming threshold (lowest-density rigid packing), but may
have substantially broader significance for condensed matter physics and
materials science.Comment: 19 pages, 5 figure
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