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Network of minima of the Thomson Problem and Smale's 7th problem.

Mehta, D. and Chen, J. and Chen, D. Z. and Kusumaatmaja, H. and Wales, D. J. (2016) 'Network of minima of the Thomson Problem and Smale's 7th problem.', Physical review letters., 117 (2). 028301.


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.

Item Type:Article
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Publisher statement:Reprinted with permission from the American Physical Society: Mehta, D., Chen, J., Chen, D. Z., Kusumaatmaja, H. & Wales, D. J. (2016). Network of Minima of the Thomson Problem and Smale's 7th Problem. Physical Review Letters, American Physical Society. 117(2), 028301 © 2016 by the American Physical Society. 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, modified, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.
Date accepted:26 May 2016
Date deposited:29 June 2016
Date of first online publication:06 July 2016
Date first made open access:No date available

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