Victor Figueroa | Searcharxiv

arXiv2025

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

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.

cond-mat.soft↗

arXiv2025

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 $N\leq24$ and estimated the growth of the number of stationary states to be exponential in $N$.

cond-mat.soft↗