A mesoscopic theory for stochastic coupled oscillators

TL;DR

This paper develops a mesoscopic framework for finite stochastic coupled oscillators, enabling precise analysis of fluctuations and synchronization transitions, advancing understanding beyond deterministic approximations.

Contribution

It introduces a general mesoscopic description for stochastic oscillator populations, providing accurate expressions for the order parameter and new insights into fluctuations and critical behavior.

Findings

Derived closed-form expressions for the stochastic Kuramoto model's order parameter.

Revealed detailed fluctuation characteristics of stochastic oscillator ensembles.

Provided new understanding of the critical exponents in synchronization transitions.

Abstract

The celebrated Ott-Antonsen ansatz for coupled oscillators provides a useful framework to work with deterministic systems in the thermodynamic limit, but remains just an approximation for stochastic models. In this paper, I construct a general mesoscopic description of finite-sized populations of stochastic coupled oscillators and apply it to study the stochastic Kuramoto model. From such a mesoscopic description it is possible to obtain the natural, multiplicative fluctuations of the oscillator ensemble. The analysis allows one to derive highly accurate, closed expressions for the stochastic Kuramoto model's order parameter for the first time. Moreover, it is possible to get novel insights into the system's fluctuations and the synchronization transition's critical exponents which were inaccessible before.