Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves

Borodachov, S. V.

Canadian journal of mathematics, Tome 64 (2012) no. 1, pp. 24-43

Résumé

We consider the problem of minimizing the energy of $N$ points repelling each other on curves in ${{\mathbb{R}}^{d}}$ with the potential ${{\left| x\,-\,y \right|}^{-s}},\,s\,\ge \,1$ , where $\left| \cdot\right|$ is the Euclidean norm. For a sufficiently smooth, simple, closed, regular curve, we find the next order term in the asymptotics of the minimal s-energy. On our way, we also prove that at least for $s\,\ge \,2$ , the minimal pairwise distance in optimal configurations asymptotically equals $L/N,\,N\to \,\infty $ , where $L$ is the length of the curve.

Détail

DOI : 10.4153/CJM-2011-038-5

Mots-clés : 31C20, 65D17, minimal discrete Riesz energy, lower order term, power law potential, separation radius

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