Estimating the Number of Stable Configurations for the Generalized Thomson Problem

Calef, Matthew T.; Griffiths, Whitney; Schulz, Alexia

doi:10.1007/s10955-015-1245-6

Cited by 10 publications

(11 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…1. In all cases [68], we have obtained more minima than previous results given in the literature [27,69]. It is known that the number of local minima for a molecular system usually increases exponentially with system size [27,70,71].…”

mentioning

confidence: 62%

“…As in previous results for Lennard-Jones clusters [75][76][77], our networks are undirected and unweighted graphs and, hence, agnostic about all other information, such as barrier heights or transition rates between the minima. "EH" refers to the estimate of the number of minima given in [27], "CGS1" is a recent [69] fit to the number of minima found in previous calculations, and "CGS2" is an estimate [69] for the number of minima (we use the maximum out of those suggested in [69]). The lines connecting data points are a guide to the eye.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Mehta

Chen

et al. 2016

Phys. Rev. Lett.

View full text Add to dashboard Cite

Readers may view, browse, and/or download material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modied, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. The Thomson problem, arrangement of identical charges on the surface of a sphere, has found many applications in physics, chemistry and biology. Here, we show that the energy landscape of the Thomson problem for N particles with N ¼ 132, 135, 138, 141, 144, 147, and 150 is single funneled, characteristic of a structure-seeking organization where the global minimum is easily accessible. Algorithmically, constructing starting points close to the global minimum of such a potential with spherical constraints is one of Smale's 18 unsolved problems in mathematics for the 21st century because it is important in the solution of univariate and bivariate random polynomial equations. By analyzing the kinetic transition networks, we show that a randomly chosen minimum is, in fact, always "close" to the global minimum in terms of the number of transition states that separate them, a characteristic of small world networks.

show abstract

mentioning

confidence: 62%

mentioning

confidence: 99%

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Mehta

Chen

et al. 2016

Phys. Rev. Lett.

View full text Add to dashboard Cite

show abstract

“…Despite the apparent simplicity, both problems provide a serious computational challenge, of increasing difficulty with N : in particular, the number of local minima of the total electrostatic energy grows very fast with N (for the case of the Thomson problem see for example the discussion in Ref. [10]). As a result, the identification of the global minimum of a system of N charges typically requires extensive numerical calculations: in the absence of a formal criterium to establish whether a given configuration of equilibrium is a global minimum, one has to repeat the numerical calculations several times, keeping N fixed, and regard the configuration with lowest energy among those obtained as a probable candidate for a global minimum.…”

mentioning

confidence: 99%

Comment on “Thomson rings in a disk”

Amore

2017

Phys. Rev. E

View full text Add to dashboard Cite

We have found that the minimum energy configuration of N=395 charges confined in a disk and interacting via the Coulomb potential, reported by Cerkaski et al. [Phys. Rev. E 91, 032312 (2015)PLEEE81539-375510.1103/PhysRevE.91.032312], is not a global minimum of the total electrostatic energy. We have identified a large number of configurations with lower energy, where defects are present close to the center of the disk; thus, the formation of a hexagonal core and valence circular rings for the centered configurations, predicted by the model of the above-mentioned reference, is not supported by numerical evidence, and the configurations obtained with this model cannot be used as a guide for the numerical calculations, as claimed by the authors.

show abstract

“…For S 2 these expressions can be used to directly evaluate the Weyl sums (16), and hence their sum of squares, and their derivatives.…”

Section: Evaluating Criteriamentioning

confidence: 99%

Efficient Spherical Designs with Good Geometric Properties

Womersley

2018

Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

View full text Add to dashboard Cite

Dedicated to Ian H. Sloan on the occasion of his 80th birthday in acknowledgement of his many fruitful ideas and generosity.Abstract Spherical t-designs on S d ⊂ R d+1 provide N nodes for an equal weight numerical integration rule which is exact for all spherical polynomials of degree at most t. This paper considers the generation of efficient, where N is comparable to (1 + t) d /d, spherical t-designs with good geometric properties as measured by their mesh ratio, the ratio of the covering radius to the packing radius. Results for S 2 include computed spherical t-designs for t = 1, ..., 180 and symmetric (antipodal) t-designs for degrees up to 325, all with low mesh ratios. These point sets provide excellent points for numerical integration on the sphere. The methods can also be used to computationally explore spherical t-designs for d = 3 and higher.

show abstract

Estimating the Number of Stable Configurations for the Generalized Thomson Problem

Cited by 10 publications

References 23 publications

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Comment on “Thomson rings in a disk”

Efficient Spherical Designs with Good Geometric Properties

Contact Info

Product

Resources

About