Network of Minima of the Thomson Problem and Smale's 7th Problem

Mehta, D.; Chen, J.; Chen, D.Z.; Kusumaatmaja, H.; Wales, D.J.

Abstract

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=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.

Citation

Mehta, D., Chen, J., Chen, D., Kusumaatmaja, H., & Wales, D. (2016). Network of Minima of the Thomson Problem and Smale's 7th Problem. Physical Review Letters, 117(2), Article 028301. https://doi.org/10.1103/physrevlett.117.028301

• Journal Article Type: Article
• Acceptance Date: May 26, 2016
• Online Publication Date: Jul 6, 2016
• Publication Date: Jul 6, 2016
• Deposit Date: Jun 28, 2016
• Publicly Available Date: Jun 29, 2016
• Journal: Physical Review Letters
• Print ISSN: 0031-9007
• Electronic ISSN: 1079-7114
• Publisher: American Physical Society
• Peer Reviewed: Peer Reviewed
• Volume: 117
• Issue: 2
• Article Number: 028301
• DOI: https://doi.org/10.1103/physrevlett.117.028301
• Public URL: https://durham-repository.worktribe.com/output/1401632
• Related Public URLs: http://arxiv.org/abs/1605.08459