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

TL;DR

This paper investigates the energy landscape of point configurations on a sphere interacting via the logarithmic potential, revealing exponential growth in local minima and stationary states up to N=160, with insights into the solution landscape for smaller N.

Contribution

It provides a detailed exploration of the energy landscape for the logarithmic potential, including the growth of local minima and stationary states, extending previous work to larger N.

Findings

Number of local minima grows exponentially with N.

Number of stationary states also grows exponentially for N ≤ 24.

Growth is similar but weaker than in the Thomson problem.

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.