Dynamics of Particles on a Curve with Pairwise Hyper-singular Repulsion

TL;DR

This paper studies the long-term behavior of particles constrained on a smooth closed curve, showing they tend to distribute uniformly and minimize hyper-singular Riesz energy over time.

Contribution

It demonstrates that particles governed by hyper-singular Riesz energy gradient flow asymptotically become uniformly distributed along the curve, regardless of initial positions.

Findings

Particles' Riesz energy approaches the minimum over time

Particles become uniformly distributed along the curve

Results hold for any number of particles and initial conditions

Abstract

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.