Dynamics of particles on a curve with pairwise hyper-singular repulsion

We investigate the large time behavior of \begin{document}$ N $\end{document} particles restricted to a smooth closed curve in \begin{document}$ \mathbb{R}^d $\end{document} and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz \begin{document}$ s $\end{document}-energy with \begin{document}$ s>1. $\end{document} We show that regardless of their initial positions, for all \begin{document}$ N $\end{document} and time \begin{document}$ t $\end{document} large, their normalized Riesz \begin{document}$ s $\end{document}-energy will be close to the \begin{document}$ N $\end{document}-point minimal possible energy. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.