Global Minima of the Thomson Problem in a Disk: A Molecular Dynamics Approach with Fixed Border Charges

Abstract

We report improved global-minimum configurations for the classical Thomson problem of $N=60$, $61$, $92$, and $99$ repulsive Coulomb charges confined to a disk. By combining the quenched molecular dynamics (QMD) method with the fixed-border heuristic introduced by Amore and Zarate, we systematically obtain configurations with energies $E_{\mathrm{QMD}}(60)=2159.3584240930$, $E_{\mathrm{QMD}}(61)=2237.19264190$, $E_{\mathrm{QMD}}(92)=5358.35353314$, and $E_{\mathrm{QMD}}(99)=6254.83029083$, which improve upon the previously best-known values. For $N=60$, the Voronoi diagram of our configuration differs from the one reported earlier. The symmetries for $N=61$ $(C_2)$ and $N=99$ $(D_1)$ are confirmed and are consistent with the known symmetry patterns for these configurations. For $N=92$, whereas the previously reported Voronoi diagram has lower symmetry, our solution exhibits a clear $D_1$ (axial) symmetry of defects. The results are highly reproducible across multiple independent runs, providing strong evidence that these configurations are robust global-minimum candidates.

10 pages, 1 figure, 1 table