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

Year: 2026
Type: preprint
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 $N=12-10^5$. 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 $N\sim10^3$, with a direct extension to spherical geometry.
DOI: 10.48550/arxiv.2609.15428
Language: English
Open Access status: green
OpenAlex ID: W7213348259
Cites: 7
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Related to: 10