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

Area of Science:

• Physics
• Chemistry
• Biology
• Mathematics

Background:

• The Thomson problem involves arranging identical charges on a sphere.
• It has diverse applications across scientific disciplines.
• Finding the global minimum energy configuration is computationally challenging.

Purpose of the Study:

• To investigate the energy landscape of the Thomson problem for specific particle numbers (N=132-150).
• To determine if the energy landscape is single-funneled, indicating easy accessibility to the global minimum.
• To analyze the proximity of random minima to the global minimum using kinetic transition networks.

Main Methods:

• Simulations of the Thomson problem for N=132, 135, 138, 141, 144, 147, and 150.
• Analysis of the energy landscape to identify funneling properties.
• Examination of kinetic transition networks to study minimum proximity.

Main Results:

• The energy landscape for the studied particle numbers is confirmed to be single-funneled.
• This suggests a structure-seeking organization where the global minimum is readily accessible.
• Randomly chosen minima are found to be close to the global minimum in the network of transitions.

Conclusions:

• The Thomson problem's energy landscape exhibits single-funneled characteristics for N=132-150.
• This implies efficient pathways to the global energy minimum.
• The findings relate to Smale's 18th problem and exhibit small-world network properties.