New upper bounds on sphere packings I

Cohn, Henry; Elkies, Noam D.

doi:10.4007/annals.2003.157.689

Cited by 285 publications

(485 citation statements)

References 22 publications

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“…Note however that, in dimensions d = 4, 8, 24, there are special lattices which are believed to solve the best packing problem [57]. Although the sphere packing problem in high dimension plays an important role in information theory, it is natural to restrict ourselves to the (physically relevant) cases of dimension d = 1, 2, 3.…”

Section: Crystallization Results and Sphere Packingmentioning

confidence: 99%

The crystallization conjecture: a review

Blanc¹,

Lewin²

2015

EMS Surv. Math. Sci.

140

191

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In this article we describe the crystallization conjecture. It states that, in appropriate physical conditions, interacting particles always place themselves into periodic configurations, breaking thereby the natural translation-invariance of the system. This famous problem is still largely open. Mathematically, it amounts to studying the minima of a real-valued function defined on R 3N where N is the number of particles, which tends to infinity. We review the existing literature and mention several related open problems, of which many have not been thoroughly studied.

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Section: Crystallization Results and Sphere Packingmentioning

confidence: 99%

The crystallization conjecture: a review

Blanc¹,

Lewin²

2015

EMS Surv. Math. Sci.

140

191

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“…Even worse, Minkowsky's bound is non-constructive, and no methods are known which would allow to construct a lattice which satisfies at least that bound in very high dimensions. Arguably the most important recent contribution in this respect has been given by the works 10,11 in which the problem is reduced, for any given dimension, to an infinite linear programming problem. The technique is powerful -in 8 and 24 dimensions the bounds are saturated by the best known packing, proving hence their global optimality-but has not yield an understanding of the problem for generic d.…”

Section: Introductionmentioning

confidence: 99%

Random perfect lattices and the sphere packing problem

Andreanov

Scardicchio

2012

Phys. Rev. E

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Motivated by the search for best lattice sphere packings in Euclidean spaces of large dimensions we study randomly generated perfect lattices in moderately large dimensions (up to d = 19 included). Perfect lattices are relevant in the solution of the problem of lattice sphere packing, because the best lattice packing is a perfect lattice and because they can be generated easily by an algorithm. Their number however grows super-exponentially with the dimension so to get an idea of their properties we propose to study a randomized version of the algorithm and to define a random ensemble with an effective temperature in a way reminiscent of a Montecarlo simulation. We therefore study the distribution of packing fractions and kissing numbers of these ensembles and show how as the temperature is decreased the best know packers are easily recovered. We find that, even at infinite temperature, the typical perfect lattices are considerably denser than known families (like A d and D d ) and we propose two hypotheses between which we cannot distinguish in this paper: one in which they improve Minkowsky's bound φ ∼ 2 −(0.84±0.06)d , and a competitor, in which their packing fraction decreases super-exponentially, namely φ ∼ d −ad but with a very small coefficient a = 0.06 ± 0.04. We also find properties of the random walk which are suggestive of a glassy system already for moderately small dimensions. We also analyze local structure of network of perfect lattices conjecturing that this is a scale-free network in all dimensions with constant scaling exponent 2.6 ± 0.1.

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“…Physicists have studied hard-sphere packings in high dimensions to gain insight into ground and glassy states of matter as well as phase behavior in lower dimensions [23][24][25][26][27][28]. The determination of the densest packings in arbitrary dimension is a problem of longstanding interest in discrete geometry and number theory [22,29,30].…”

Section: Introductionmentioning

confidence: 99%

Robust algorithm to generate a diverse class of dense disordered and ordered sphere packings via linear programming

2010

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We have formulated the problem of generating dense packings of nonoverlapping, non-tiling nonspherical particles within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem called the Adaptive Shrinking Cell (ASC) formulation [S. Torquato and Y. Jiao, Phys. Rev. E 80, 041104 (2009)]. Because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in d-dimensionalEuclidean space R d , 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 R d for d = 2, 3, 4, 5 and 6 with a diversity of disorder and densities up to the respective maximal densities. A novel feature of this deterministic algorithm is that it can produce a broad range of inherent structures (locally maximally dense and mechanically stable packings), besides the usual disordered ones (such as the maximally random jammed state), with very small computational cost compared to that of the best known packing algorithms by tuning the radius of the influence sphere. For example, 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 [0.6, 0.7408 . . .], where π/ √ 18 = 0.7408 . . .corresponds to the density of the densest packing. We also apply the algorithm to generate various disordered packings as well as the maximally dense packings for d = 2, 4, 5 and 6. Our jammed sphere packings are characterized and compared to the corresponding packings generated by the well-known Lubachevsky-Stillinger molecular-dynamics packing algorithm. 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.

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New upper bounds on sphere packings I

Cited by 285 publications

References 22 publications

The crystallization conjecture: a review

The crystallization conjecture: a review

Random perfect lattices and the sphere packing problem

Robust algorithm to generate a diverse class of dense disordered and ordered sphere packings via linear programming

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