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