9,916 research outputs found
New upper bounds on sphere packings I
We develop an analogue for sphere packing of the linear programming bounds
for error-correcting codes, and use it to prove upper bounds for the density of
sphere packings, which are the best bounds known at least for dimensions 4
through 36. We conjecture that our approach can be used to solve the sphere
packing problem in dimensions 8 and 24.Comment: 26 pages, 1 figur
Thurston's sphere packings on 3-dimensional manifolds, I
Thurston's sphere packing on a 3-dimensional manifold is a generalization of
Thusrton's circle packing on a surface, the rigidity of which has been open for
many years. In this paper, we prove that Thurston's Euclidean sphere packing is
locally determined by combinatorial scalar curvature up to scaling, which
generalizes Cooper-Rivin-Glickenstein's local rigidity for tangential sphere
packing on 3-dimensional manifolds. We also prove the infinitesimal rigidity
that Thurston's Euclidean sphere packing can not be deformed (except by
scaling) while keeping the combinatorial Ricci curvature fixed.Comment: Arguments are simplife
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Deriving Finite Sphere Packings
Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for deriving all packings of n spheres in R3 satisfying minimal rigidity constraints (≥ 3 contacts per sphere and ≥ 3n − 6 total contacts). We derive such packings for n ≤ 10 and provide a preliminary set of maximum contact packings for 10 < n ≤ 20. The resultant set of packings has some striking features; among them are the following: (i) all minimally rigid packings for n ≤ 9 have exactly 3n−6 contacts; (ii) nonrigid packings satisfying minimal rigidity constraints arise for n ≥ 9; (iii) the number of ground states (i.e., packings with the maximum number of contacts) oscillates with respect to n; (iv) for 10 ≤ n ≤ 20 there are only a small number of packings with the maximum number of contacts, and for 10 ≤ n < 13 these are all commensurate with the hexagonal close-packed lattice. The general method presented here may have applications to other related problems in mathematics, such as the Erdos repeated distance problem and Euclidean distance matrix completion problems.Engineering and Applied SciencesPhysic
Dense and Nearly Jammed Random Packings of Freely Jointed Chains of Tangent Hard Spheres
Dense packings of freely jointed chains of tangent hard spheres are produced by a novel Monte Carlo method. Within statistical uncertainty, chains reach a maximally random jammed (MRJ) state at the same volume fraction as packings of single hard spheres. A structural analysis shows that as the MRJ state is approached (i) the radial distribution function for chains remains distinct from but approaches that of single hard sphere packings quite closely, (ii) chains undergo progressive collapse, and (iii) a small but increasing fraction of sites possess highly ordered first coordination shells
High-Performance Computing of Flow, Diffusion, and Hydrodynamic Dispersion in Random Sphere Packings
This thesis is dedicated to the study of mass transport processes (flow, diffusion, and hydrodynamic dispersion) in computer-generated random sphere packings. Periodic and confined packings of hard impermeable spheres were generated using Jodrey–Tory and Monte Carlo procedure-based algorithms, mass transport in the packing void space was simulated using the lattice Boltzmann and random walk particle tracking methods. Simulation codes written in C programming language using MPI library allowed an efficient use of the high-performance computing systems (supercomputers).
The first part of this thesis investigates the influence of the cross-sectional geometry of the confined random sphere packings on the hydrodynamic dispersion. Packings with different values of porosity (interstitial void space fraction) generated in containers of circular, quadratic, rectangular, trapezoidal, and irregular (reconstructed) geometries were studied, and resulting pre-asymptotic and close-to-asymptotic hydrodynamic dispersion coefficients were analyzed. It was demonstrated i) a significant impact of the cross-sectional geometry and porosity on the hydrodynamic dispersion coefficients, and ii) reduction of the symmetry of the cross section results in longer times to reach close-to-asymptotic values and larger absolute values of the hydrodynamic dispersion coefficients. In case of reconstructed geometry, good agreement with experimental data was found. In the second part of this thesis i) length scales of heterogeneity persistent in unconfined and confined sphere packings were analyzed and correlated with a time behavior of the hydrodynamic dispersion coefficients; close-to-asymptotic values of the dispersion coefficients (expressed in terms of plate height) were successfully fitted to the generalized Giddings equation; ii) influence of the packing microstructural disorder on the effective diffusion and hydrodynamic dispersion coefficients was investigated and clear qualitative corellation with geometrical descriptors (which are based on Delaunay and Voronoi spatial tessellations) was demonstrated
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
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
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