Finite-size effects in a stochastic Kuramoto model

TL;DR

This paper introduces a collective coordinate method to analyze finite-size stochastic Kuramoto oscillators, revealing that their mean phase undergoes Brownian diffusion with variance inversely proportional to the ensemble size.

Contribution

The study develops a novel collective coordinate framework that accurately captures finite-size effects in stochastic Kuramoto models, including phase diffusion.

Findings

Mean phase exhibits Brownian diffusion in finite ensembles.

Variance of phase diffusion scales as 1/N.

Method accurately predicts finite-size effects.

Abstract

We present a collective coordinate approach to study the collective behaviour of a finite ensemble of $N$ stochastic Kuramoto oscillators using two degrees of freedom; one describing the shape dynamics of the oscillators and one describing their mean phase. Contrary to the thermodynamic limit $N\to\infty$ in which the mean phase of the cluster of globally synchronized oscillators is constant in time, the mean phase of a finite-size cluster experiences Brownian diffusion with a variance proportional to $1/N$. This finite-size effect is quantitatively well captured by our collective coordinate approach.