arXiv subjects

Georgiy K. Lavrov

Publications and source records attributed to Georgiy K. Lavrov.

3 recordsLinked to original sources

arXiv2026

Lowest-known Energy Configuration of $N=100\,000$ Coulomb Charges in a Disk: Breaking the $10^{5}$ Barrier

We report the first calculation of the lowest-known energy configuration for $N=100\,000$ classical point charges confined to a disk and interacting via the $1/r$ Coulomb potential - a system size never before achieved for the Thomson problem in a disk, more than doubling the previous record of $N=40\,886$ reported by Amore and Zarate. An adaptive defect-targeting subdomain optimization strategy yields an approximately linear growth of the wall-clock time per cycle with $N$; for $N=100\,000$ a single $24$-core CPU workstation without GPU acceleration required $\approx31$ hours. The energy $E_{\min}=7.80466624157\times10^{9}$ deviates from the standard empirical asymptotic expansion by only $2.525\times10^{-7}$. We demonstrate that this remarkable agreement provides a stringent test of the rigorous next-order asymptotic theory for Riesz interactions, while the residual deviation is consistent with an $O(N^{4/3})$ boundary-layer correction whose absence from the standard fitting basis biases the empirical $N^{3/2}$ coefficient away from its exact crystalline value, thereby tying the deviation to the two-dimensional crystallization conjecture. Bond-orientational order analysis reveals a polycrystalline bulk threaded by radial grain boundaries, establishing a new benchmark for two-dimensional Coulomb systems.

cond-mat.stat-mech↗

arXiv2026

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

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.

cond-mat.stat-mech↗

arXiv2026

Shannon Entropy as an Order Parameter for the Two-Dimensional Confined Coulomb Systems: Exact Balance Law, Topological Charge Sum Rule, and Boundary Saturation

We establish a complete information-theoretic framework for the known minimum-energy configurations of the two-dimensional Thomson problem on a hard-wall disk, based on the Shannon entropy of the Voronoi topological charge distribution. Two exact results are proven and hold for all studied systems of size $N=12$-$10^5$. First, an entropy balance law decomposes the total entropy into bulk, boundary, and mixing contributions without approximation. Second, a topological charge sum rule fixes the total charge at twice the number of boundary particles plus six; together with the observed separation of bulk and boundary coordination numbers, it yields exact relations between the populations of three- and four-coordinated boundary particles and the bulk charge. We further prove that the boundary entropy is bounded by the binary maximum and that saturation of this bound is equivalent to the bulk charge per boundary particle approaching minus one half, a regime that the framework itself restricts to boundaries of at least twelve particles. All identities are verified numerically to machine precision, locating bulk nucleation at $56$ particles, the maximum of global disorder at $146$, and the onset of boundary saturation beyond about $1000$. The framework carries over to closed spherical geometry, where the total charge is fixed at twelve.

cond-mat.stat-mech↗