The ground state for sticky disks

Heitmann, Raymond C.; Radin, Charles

doi:10.1007/bf01014644

Cited by 96 publications

(118 citation statements)

References 4 publications

Supporting

Mentioning

117

Contrasting

Order By: Relevance

“…The fact that FCC is the unique minimizer among Bravais lattices was proved by Gauss [106]. The link between the crystallization problem and the sphere packing problem has been highlighted by Heitmann and Radin in [129]. Indeed, if the interaction potential V is given by…”

Section: Crystallization Results and Sphere Packingmentioning

confidence: 99%

See 1 more Smart Citation

The crystallization conjecture: a review

Blanc¹,

Lewin²

2015

EMS Surv. Math. Sci.

134

191

View full text Add to dashboard Cite

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.

show abstract

Section: Crystallization Results and Sphere Packingmentioning

confidence: 99%

“…It has been proposed by Wulff [249], and proved rigorously for a hard sphere model 4 in dimension two by Au Yeung, Friesecke and Schmidt in [17,220]. This work is based on results by Radin et al [194,129]. We refer for instance to [29,28,52] for similar results on the Ising model.…”

Section: Vortices and Crystallization In Dimensionmentioning

confidence: 87%

The crystallization conjecture: a review

Blanc¹,

Lewin²

2015

EMS Surv. Math. Sci.

134

191

View full text Add to dashboard Cite

show abstract

“…The fundamental understanding of these phenomena is still in a primitive stage, mostly based on variational computations and special assumptions on the structure of the low-energy states. There are just a few special cases where the periodicity of the ground state can be proved from first principles [13,25,26,40,41].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Periodic Striped Ground States in Ising Models with Competing Interactions

Giuliani

Seiringer

2016

Commun. Math. Phys.

View full text Add to dashboard Cite

Abstract:We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent p of the long range interaction is larger than d + 1, with d the space dimension, this happens for all values of J smaller than a critical value J c ( p), beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for p > 2d and J in a left neighborhood of J c ( p). In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes (d = 2) or slabs (d = 3), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity.

show abstract

“…The important crystallization result we are concerned with is due to Heitmann and Radin [16]. One starts from the prototypical Lennard-Jones energy for a system of N identical atoms with positions x 1 , .., x N ∈ R d , (1.1)…”

Section: Introductionmentioning

confidence: 99%

“…We show that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems. In particular, we give a simplified proof of the Heitmann-Radin crystallization theorem [16], which concerns a system of N identical atoms in two dimensions interacting via the idealized pair potential V (r) = +∞ if r < 1, −1 if r = 1, 0 if r > 1. This is done by endowing the bond graph of a general particle configuration with a suitable notion of discrete curvature, and appealing to a discrete GaussBonnet theorem [18] which, as its continuous cousins, relates the sum/integral of the curvature to topological invariants.…”

mentioning

confidence: 99%

Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Luca

Friesecke

2017

J Nonlinear Sci

View full text Add to dashboard Cite

Abstract. We show that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems. In particular, we give a simplified proof of the Heitmann-Radin crystallization theorem [16], which concerns a system of N identical atoms in two dimensions interacting via the idealized pair potential V (r) = +∞ if r < 1, −1 if r = 1, 0 if r > 1. This is done by endowing the bond graph of a general particle configuration with a suitable notion of discrete curvature, and appealing to a discrete GaussBonnet theorem [18] which, as its continuous cousins, relates the sum/integral of the curvature to topological invariants. This leads to an exact geometric decomposition of the Heitmann-Radin energy into (i) a combinatorial bulk term, (ii) a combinatorial perimeter, (iii) a multiple of the Euler characteristic, and (iv) a natural topological energy contribution due to defects. An analogous exact geometric decomposition is also established for soft potentials such as the Lennard-Jones potential V (r) = r −6 − 2r −12 , where two additional contributions arise, (v) elastic energy and (vi) energy due to non-bonded interactions.

show abstract

The ground state for sticky disks

Cited by 96 publications

References 4 publications

The crystallization conjecture: a review

The crystallization conjecture: a review

Periodic Striped Ground States in Ising Models with Competing Interactions

Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Contact Info

Product

Resources

About