Mean-field approach to finite-size fluctuations in the Kuramoto-Sakaguchi model

TL;DR

This paper introduces a new analytical approach to quantify finite-size fluctuations in the Kuramoto-Sakaguchi model, providing explicit formulas and validating them with simulations.

Contribution

It offers an ab initio method to analytically describe fluctuations without prior assumptions, applicable to other particle systems.

Findings

Derived explicit covariance functions for fluctuations.

Calculated variance of the order parameter magnitude.

Validated analytical results with numerical simulations.

Abstract

We develop an ab initio approach to describe the statistical behavior of finite-size fluctuations in the deterministic Kuramoto-Sakaguchi model. We obtain explicit expressions for the covariance function of fluctuations of the complex order parameter and determine the variance of its magnitude entirely in terms of the equation parameters. Our results rely on an explicit complex-valued formula for the solution of the Adler equation. We present analytical results for both the sub- and the super-critical case. Moreover, our framework does not require any prior knowledge about the structure of the partially synchronized state. We corroborate our results with numerical simulations of the full Kuramoto-Sakaguchi model. The proposed methodology is sufficiently general such that it can be applied to other interacting particle systems.