Kinetic Transition Networks for the Thomson Problem and Smale's 7th Problem

TL;DR

This paper analyzes the energy landscape of the Thomson Problem for specific particle counts, revealing a single-funnel structure and small-world properties that facilitate finding global minima, with implications for Smale's 7th problem.

Contribution

It demonstrates that the energy landscape for certain particle numbers is single-funnelled and exhibits small-world characteristics, aiding in solving Smale's 7th problem.

Findings

Energy landscape is single-funnelled for N=132 to 150.

Random minima are close to the global minimum in transition states.

Energy landscape exhibits small-world network properties.

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 funnelled, characteristic of a structure-seeking organisation 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 analysing 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.