Papers › Classifying minimum energy states for interacting particles: Spherical Shells
Classifying minimum energy states for interacting particles: Spherical Shells
Cameron Davies, Tongseok Lim, Robert J. McCann
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
Particles interacting through long-range attraction and short-range repulsion given by power-laws have been widely used to model physical and biological systems, and to predict or explain many of the patterns they display. Apart from rare values of the attractive and repulsive exponents (α,β), the energy minimizing configurations of particles are not explicitly known, although simulations and local stability considerations have led to conjectures with strong evidence over a much wider region of parameters. For a segment β=2<α<4 on the mildly repulsive frontier we employ strict convexity to conclude that the energy is uniquely minimized (up to translation) by a spherical shell. In a companion work, we show that in the mildly repulsive range α>β≥2, a unimodal threshold 2<α_(Δⁿ)(β) ≤max{β,4} exists such that equidistribution of particles over a unit diameter regular n-simplex minimizes the energy if and only if α≥α_(Δⁿ)(β) (and minimizes uniquely up to rigid motions if strict inequality holds). At the point (α,β)=(2,4) separating these regimes, we show the minimizers all lie on a sphere and are precisely characterized by sharing all first and second moments with the spherical shell. Although the minimizers need not be asymptotically stable, our approach establishes d_α-Lyapunov nonlinear stability of the associated (d₂-gradient) aggregation dynamics near the minimizer in both of these adjacent regimes -- without reference to linearization. The L^α-Kantorovich-Rubinstein distance d_α which quantifies stability is chosen to match the attraction exponent.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections