Shannon Entropy as an Order Parameter for the Two-Dimensional Confined Coulomb Systems: Exact Balance Law, Topological Charge Sum Rule, and Boundary Saturation
Abstract
We establish a complete information-theoretic framework for the 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 identities are proven and verified to machine precision for all studied systems of size N=12−105. 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 disjointness of bulk and boundary coordination sets − justified analytically via the Delaunay empty-circumcircle criterion and elastic disclination splitting − it relates the three- and four-coordinated boundary populations directly to the bulk charge. Applying Euler's theorem to the outermost Delaunay annulus yields an exact shell-mismatch identity: the four-coordinated boundary population equals the adjacent interior shell population, whereas the three-coordinated population equals their step mismatch, directly detecting third-shell nucleation at N=30. Furthermore, we prove that the boundary entropy is bounded by ln2 and that its saturation is equivalent to the bulk charge per boundary particle approaching −1/2, a regime restricted self-consistently to boundaries of at least $12$ particles. Numerically, we locate bulk nucleation at N=56, maximal global disorder at N=146, and boundary saturation above N∼103, with a direct extension to spherical geometry.
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BibTeX
@misc{lavrov2026shannon,
title = {{Shannon Entropy as an Order Parameter for the Two-Dimensional Confined Coulomb Systems: Exact Balance Law, Topological Charge Sum Rule, and Boundary Saturation}},
author = {Georgiy K. Lavrov and Eduard G. Nikonov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.15428},
eprint = {2609.15428},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15428},
}
RIS
TY - UNPB
TI - Shannon Entropy as an Order Parameter for the Two-Dimensional Confined Coulomb Systems: Exact Balance Law, Topological Charge Sum Rule, and Boundary Saturation
AU - Georgiy K. Lavrov
AU - Eduard G. Nikonov
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2609.15428
UR - https://arxiv.org/abs/2609.15428
AB - We establish a complete information-theoretic framework for the 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 identities are proven and verified to machine precision 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 disjointness of bulk and boundary coordination sets − justified analytically via the Delaunay empty-circumcircle criterion and elastic disclination splitting − it relates the three- and four-coordinated boundary populations directly to the bulk charge. Applying Euler's theorem to the outermost Delaunay annulus yields an exact shell-mismatch identity: the four-coordinated boundary population equals the adjacent interior shell population, whereas the three-coordinated population equals their step mismatch, directly detecting third-shell nucleation at N=30. Furthermore, we prove that the boundary entropy is bounded by ln2 and that its saturation is equivalent to the bulk charge per boundary particle approaching −1/2, a regime restricted self-consistently to boundaries of at least $12$ particles. Numerically, we locate bulk nucleation at N=56, maximal global disorder at N=146, and boundary saturation above N∼10^3, with a direct extension to spherical geometry.
ER -