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

Abstract

<p style='text-indent:20px;'>We investigate the large time behavior of <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> particles restricted to a smooth closed curve in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^d $\end{document}</tex-math></inline-formula> and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz <inline-formula><tex-math id="M3">\begin{document}$ s $\end{document}</tex-math></inline-formula>-energy with <inline-formula><tex-math id="M4">\begin{document}$ s&gt;1. $\end{document}</tex-math></inline-formula> We show that regardless of their initial positions, for all <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula> and time <inline-formula><tex-math id="M6">\begin{document}$ t $\end{document}</tex-math></inline-formula> large, their normalized Riesz <inline-formula><tex-math id="M7">\begin{document}$ s $\end{document}</tex-math></inline-formula>-energy will be close to the <inline-formula><tex-math id="M8">\begin{document}$ N $\end{document}</tex-math></inline-formula>-point minimal possible energy. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.</p>