Exploring the Energy Landscape of the Thomson Problem: Local Minima and Stationary States

We conducted a comprehensive numerical investigation of the energy landscape of the Thomson problem for systems up to \(N=150\). Our results show the number of distinct configurations grows exponentially with N, but significantly faster than previously reported. Furthermore, we find that the average energy gap between independent configurations at a given N decays exponentially with N, dramatically increasing the computational complexity for larger systems. Finally, we developed a novel approach that reformulates the search for stationary points in the Thomson problem (or similar systems) as an equivalent minimization problem using a specifically designed potential. Leveraging this method, we performed a detailed exploration of the solution landscape for \(N\le 24\) and estimated the growth of the number of stationary states to be exponential in N.