193 research outputs found
Contact numbers for congruent sphere packings in Euclidean 3-space
Continuing the investigations of Harborth (1974) and the author (2002) we
study the following two rather basic problems on sphere packings. Recall that
the contact graph of an arbitrary finite packing of unit balls (i.e., of an
arbitrary finite family of non-overlapping unit balls) in Euclidean 3-space is
the (simple) graph whose vertices correspond to the packing elements and whose
two vertices are connected by an edge if the corresponding two packing elements
touch each other. One of the most basic questions on contact graphs is to find
the maximum number of edges that a contact graph of a packing of n unit balls
can have in Euclidean 3-space. Our method for finding lower and upper estimates
for the largest contact numbers is a combination of analytic and combinatorial
ideas and it is also based on some recent results on sphere packings. Finally,
we are interested also in the following more special version of the above
problem. Namely, let us imagine that we are given a lattice unit sphere packing
with the center points forming the lattice L in Euclidean 3-space (and with
certain pairs of unit balls touching each other) and then let us generate
packings of n unit balls such that each and every center of the n unit balls is
chosen from L. Just as in the general case we are interested in finding good
estimates for the largest contact number of the packings of n unit balls
obtained in this way.Comment: 18 page
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
Basic Understanding of Condensed Phases of Matter via Packing Models
Packing problems have been a source of fascination for millenia and their
study has produced a rich literature that spans numerous disciplines.
Investigations of hard-particle packing models have provided basic insights
into the structure and bulk properties of condensed phases of matter, including
low-temperature states (e.g., molecular and colloidal liquids, crystals and
glasses), multiphase heterogeneous media, granular media, and biological
systems. The densest packings are of great interest in pure mathematics,
including discrete geometry and number theory. This perspective reviews
pertinent theoretical and computational literature concerning the equilibrium,
metastable and nonequilibrium packings of hard-particle packings in various
Euclidean space dimensions. In the case of jammed packings, emphasis will be
placed on the "geometric-structure" approach, which provides a powerful and
unified means to quantitatively characterize individual packings via jamming
categories and "order" maps. It incorporates extremal jammed states, including
the densest packings, maximally random jammed states, and lowest-density jammed
structures. Packings of identical spheres, spheres with a size distribution,
and nonspherical particles are also surveyed. We close this review by
identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal
of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298
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
Dense periodic packings of tori
Dense packings of nonoverlapping bodies in three-dimensional Euclidean space
are useful models of the structure of a variety of many-particle systems that
arise in the physical and biological sciences. Here we investigate the packing
behavior of congruent ring tori, which are multiply connected nonconvex bodies
of genus 1, as well as horn and spindle tori. We analytically construct a
family of dense periodic packings of unlinked tori guided by the organizing
principles originally devised for simply connected solid bodies [Torquato and
Jiao, PRE 86, 011102 (2012)]. We find that the horn tori as well as certain
spindle and ring tori can achieve a packing density higher than the densest
known packing of both sphere and ellipsoids. In addition, we study dense
packings of cluster of pair-linked ring tori (i.e., Hopf links).Comment: 15 pages, 7 figure
Highly saturated packings and reduced coverings
We introduce and study certain notions which might serve as substitutes for
maximum density packings and minimum density coverings. A body is a compact
connected set which is the closure of its interior. A packing with
congruent replicas of a body is -saturated if no members of it can
be replaced with replicas of , and it is completely saturated if it is
-saturated for each . Similarly, a covering with congruent
replicas of a body is -reduced if no members of it can be replaced
by replicas of without uncovering a portion of the space, and it is
completely reduced if it is -reduced for each . We prove that every
body in -dimensional Euclidean or hyperbolic space admits both an
-saturated packing and an -reduced covering with replicas of . Under
some assumptions on (somewhat weaker than convexity),
we prove the existence of completely saturated packings and completely reduced
coverings, but in general, the problem of existence of completely saturated
packings and completely reduced coverings remains unsolved. Also, we
investigate some problems related to the the densities of -saturated
packings and -reduced coverings. Among other things, we prove that there
exists an upper bound for the density of a -reduced covering of
with congruent balls, and we produce some density bounds for the
-saturated packings and -reduced coverings of the plane with congruent
circles
Optimal Packings of Superballs
Dense hard-particle packings are intimately related to the structure of
low-temperature phases of matter and are useful models of heterogeneous
materials and granular media. Most studies of the densest packings in three
dimensions have considered spherical shapes, and it is only more recently that
nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs
(whose shapes are defined by |x1|^2p + |x2|^2p + |x3|^2p <= 1) provide a
versatile family of convex particles (p >= 0.5) with both cubic- and
octahedral-like shapes as well as concave particles (0 < p < 0.5) with
octahedral-like shapes. In this paper, we provide analytical constructions for
the densest known superball packings for all convex and concave cases. The
candidate maximally dense packings are certain families of Bravais lattice
packings. The maximal packing density as a function of p is nonanalytic at the
sphere-point (p = 1) and increases dramatically as p moves away from unity. The
packing characteristics determined by the broken rotational symmetry of
superballs are similar to but richer than their two-dimensional "superdisk"
counterparts, and are distinctly different from that of ellipsoid packings. Our
candidate optimal superball packings provide a starting point to quantify the
equilibrium phase behavior of superball systems, which should deepen our
understanding of the statistical thermodynamics of nonspherical-particle
systems.Comment: 28 pages, 16 figure
Estimates of the optimal density and kissing number of sphere packings in high dimensions
The problem of finding the asymptotic behavior of the maximal density of
sphere packings in high Euclidean dimensions is one of the most fascinating and
challenging problems in discrete geometry. One century ago, Minkowski obtained
a rigorous lower bound that is controlled asymptotically by , where
is the Euclidean space dimension. An indication of the difficulty of the
problem can be garnered from the fact that exponential improvement of
Minkowski's bound has proved to be elusive, even though existing upper bounds
suggest that such improvement should be possible. Using a
statistical-mechanical procedure to optimize the density associated with a
"test" pair correlation function and a conjecture concerning the existence of
disordered sphere packings [S. Torquato and F. H. Stillinger, Experimental
Math. {\bf 15}, 307 (2006)], the putative exponential improvement was found
with an asymptotic behavior controlled by . Using the same
methods, we investigate whether this exponential improvement can be further
improved by exploring other test pair correlation functions correponding to
disordered packings. We demonstrate that there are simpler test functions that
lead to the same asymptotic result. More importantly, we show that there is a
wide class of test functions that lead to precisely the same exponential
improvement and therefore the asymptotic form is much
more general than previously surmised.Comment: 23 pages, 4 figures, submitted to Phys. Rev.
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