title: Kinetic Transition Networks for the Thomson Problem and Smale's Seventh Problem.
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.
authors.affiliation: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana 46556, USA.|Centre for the Subatomic Structure of Matter, Department of Physics, School of Physical Sciences, University of Adelaide, Adelaide, South Australia 5005, Australia.|Department of Chemical and Biomolecular Engineering, University of Notre Dame, Notre Dame, Indiana 46556, USA.
authors.forename: Dhagash
authors.initials: D
authors.lastname: Mehta
authors.affiliation: Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, Indiana 46556, USA.
authors.forename: Jianxu
authors.initials: J
authors.lastname: Chen
authors.affiliation: Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, Indiana 46556, USA.
authors.forename: Danny Z
authors.initials: DZ
authors.lastname: Chen
authors.affiliation: Department of Physics, Durham University, South Road, Durham DH1 3LE, United Kingdom.
authors.forename: Halim
authors.initials: H
authors.lastname: Kusumaatmaja
authors.affiliation: University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, United Kingdom.
authors.forename: David J
authors.initials: DJ
authors.lastname: Wales