767 research outputs found
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
Unexpected Density Fluctuations in Jammed Disordered Sphere Packings
We computationally study jammed disordered hard-sphere packings as large as a
million particles. We show that the packings are saturated and hyperuniform,
i.e., that local density fluctuations grow only as a logarithmically-augmented
surface area rather than the volume of the window. The structure factor shows
an unusual non-analytic linear dependence near the origin, . In
addition to exponentially damped oscillations seen in liquids, this implies a
weak power-law tail in the total correlation function, , and a
long-ranged direct correlation function.Comment: Submitted for publicatio
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
Packing Hyperspheres in High-Dimensional Euclidean Spaces
We present the first study of disordered jammed hard-sphere packings in
four-, five- and six-dimensional Euclidean spaces. Using a collision-driven
packing generation algorithm, we obtain the first estimates for the packing
fractions of the maximally random jammed (MRJ) states for space dimensions
, 5 and 6 to be , 0.31 and 0.20, respectively. To
a good approximation, the MRJ density obeys the scaling form , where and , which appears to be
consistent with high-dimensional asymptotic limit, albeit with different
coefficients. Calculations of the pair correlation function and
structure factor for these states show that short-range ordering
appreciably decreases with increasing dimension, consistent with a recently
proposed ``decorrelation principle,'' which, among othe things, states that
unconstrained correlations diminish as the dimension increases and vanish
entirely in the limit . As in three dimensions (where ), the packings show no signs of crystallization, are isostatic,
and have a power-law divergence in at contact with power-law
exponent . Across dimensions, the cumulative number of neighbors
equals the kissing number of the conjectured densest packing close to where
has its first minimum. We obtain estimates for the freezing and
melting desnities for the equilibrium hard-sphere fluid-solid transition,
and , respectively, for , and
and , respectively, for .Comment: 28 pages, 9 figures. To appear in Physical Review
Geometrical Ambiguity of Pair Statistics. I. Point Configurations
Point configurations have been widely used as model systems in condensed
matter physics, materials science and biology. Statistical descriptors such as
the -body distribution function is usually employed to characterize
the point configurations, among which the most extensively used is the pair
distribution function . An intriguing inverse problem of practical
importance that has been receiving considerable attention is the degree to
which a point configuration can be reconstructed from the pair distribution
function of a target configuration. Although it is known that the pair-distance
information contained in is in general insufficient to uniquely determine
a point configuration, this concept does not seem to be widely appreciated and
general claims of uniqueness of the reconstructions using pair information have
been made based on numerical studies. In this paper, we introduce the idea of
the distance space, called the space. The pair distances of a
specific point configuration are then represented by a single point in the
space. We derive the conditions on the pair distances that can be
associated with a point configuration, which are equivalent to the
realizability conditions of the pair distribution function . Moreover, we
derive the conditions on the pair distances that can be assembled into distinct
configurations. These conditions define a bounded region in the
space. By explicitly constructing a variety of degenerate point configurations
using the space, we show that pair information is indeed
insufficient to uniquely determine the configuration in general. We also
discuss several important problems in statistical physics based on the
space.Comment: 28 pages, 8 figure
Application of Edwards' statistical mechanics to high dimensional jammed sphere packings
The isostatic jamming limit of frictionless spherical particles from Edwards'
statistical mechanics [Song \emph{et al.}, Nature (London) {\bf 453}, 629
(2008)] is generalized to arbitrary dimension using a liquid-state
description. The asymptotic high-dimensional behavior of the self-consistent
relation is obtained by saddle-point evaluation and checked numerically. The
resulting random close packing density scaling is
consistent with that of other approaches, such as replica theory and density
functional theory. The validity of various structural approximations is
assessed by comparing with three- to six-dimensional isostatic packings
obtained from simulations. These numerical results support a growing accuracy
of the theoretical approach with dimension. The approach could thus serve as a
starting point to obtain a geometrical understanding of the higher-order
correlations present in jammed packings.Comment: 13 pages, 7 figure
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
Nucleation-induced transition to collective motion in active systems
While the existence of polar ordered states in active systems is well
established, the dynamics of the self-assembly processes are still elusive. We
study a lattice gas model of self-propelled elongated particles interacting
through excluded volume and alignment interactions, which shows a phase
transition from an isotropic to a polar ordered state. By analyzing the
ordering process we find that the transition is driven by the formation of a
critical nucleation cluster and a subsequent coarsening process. Moreover, the
time to establish a polar ordered state shows a power-law divergence
Random Sequential Addition of Hard Spheres in High Euclidean Dimensions
Employing numerical and theoretical methods, we investigate the structural
characteristics of random sequential addition (RSA) of congruent spheres in
-dimensional Euclidean space in the infinite-time or
saturation limit for the first six space dimensions ().
Specifically, we determine the saturation density, pair correlation function,
cumulative coordination number and the structure factor in each =of these
dimensions. We find that for , the saturation density
scales with dimension as , where and
. We also show analytically that the same density scaling
persists in the high-dimensional limit, albeit with different coefficients. A
byproduct of this high-dimensional analysis is a relatively sharp lower bound
on the saturation density for any given by , where is the structure factor at
(i.e., infinite-wavelength number variance) in the high-dimensional limit.
Consistent with the recent "decorrelation principle," we find that pair
correlations markedly diminish as the space dimension increases up to six. Our
work has implications for the possible existence of disordered classical ground
states for some continuous potentials in sufficiently high dimensions.Comment: 38 pages, 9 figures, 4 table
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