Exploring the energy landscape of the logarithmic potential: local minima and stationary states

Abstract

We have performed a detailed exploration of the energy landscape for configurations of points on the sphere, interacting via the logarithmic potential, and corresponding to local minima of the total energy, up to $N = 160$. The growth of $N_{\rm conf}$ (number of distinct configurations) is exponential, as for the Thomson problem, although weaker. Using the techniques described in our previous paper~\cite{Amore25} we have also explored the solution landscape of this problem for $N \leq 24$, and found that the number of stationary states is growing exponentially.