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

TL;DR

This work extends the landscape analysis from the Coulomb (Thomson) problem to point configurations on the sphere interacting via the logarithmic potential, treated as the $s \to 0$ limit of $V = 1/r^{s}$. Using a refined bouncing method that combines upgrade/downgrade steps, the authors systematically enumerate local minima up to $N = 160$ and, for $N \le 24$, explore the solution landscape of stationary states. They find exponential growth in both local minima ($N_{\rm conf}$) and stationary states, with the log-potential case exhibiting fewer independent configurations than the Coulomb case; energy gaps decay exponentially while the energy span grows linearly, and the distributions of energies and gaps follow Burr XII and Weibull forms, respectively. The analysis uncovers intricate defect structures via Voronoi topology, occasional strong overlap between log and Coulomb landscapes, and practical implications for Smale’s seventh problem, while noting the need for scalable approaches for larger $N$. Data supporting the results are available publicly, enabling further exploration and benchmarking.

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.

Figures (17)

• Figure 1: $\tilde{N}_0/N_0$ vs $N$ (blue curve) and $N_0/N_1$ (red curve).
• Figure 2: Number of local minima for the Thomson problem. The fit was done for values in the range $100 \leq N \leq 160$.
• Figure 3: Smallest energy gap for configurations with $70\leq N \leq 160$.
• Figure 4: Energy span defined as $E_{\rm max}(N) -E_{\rm min}(N)$ for configurations with $70\leq N \leq 160$.
• Figure 5: Smallest $\Delta$ and $d_E$ for $20\leq N \leq 160$.