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

http://dx.doi.org/10.4153/CJM-2011-038-5
Canad. J. Math. 64(2012), 24-43

  Published:2011-09-15
 Printed: Feb 2012

We consider the problem of minimizing the energy of $N$ points repelling each other on curves in $\mathbb{R}^d$ with the potential $|x-y|^{-s}$, $s\geq 1$, where $|\, \cdot\, |$ 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 $s\geq 2$, the minimal pairwise distance in optimal configurations asymptotically equals $L/N$, $N\to\infty$, where $L$ is the length of the curve.