Exact correlations in the nonequilibrium stationary state of the noisy Kuramoto model

TL;DR

This paper derives exact autocorrelation functions in the nonequilibrium stationary state of the noisy Kuramoto model, revealing exponential decay behavior across phases and at criticality, using a novel mapping approach.

Contribution

It introduces an exact mapping method to analyze autocorrelations in the noisy Kuramoto model and related models, providing new analytical results for their stationary states.

Findings

Autocorrelation decays exponentially in all phases and at the critical point.

Decay rate increases continuously with noise strength.

Results apply to both the Kuramoto and Brownian mean-field models.

Abstract

We obtain exact results on autocorrelation of the order parameter in the nonequilibrium stationary state of a paradigmatic model of spontaneous collective synchronization, the Kuramoto model of coupled oscillators, evolving in presence of Gaussian, white noise. The method relies on an exact mapping of the stationary-state dynamics of the model in the thermodynamic limit to the noisy dynamics of a single, non-uniform oscillator, and allows to obtain besides the Kuramoto model the autocorrelation in the equilibrium stationary state of a related model of long-range interactions, the Brownian mean-field model. Both the models show a phase transition between a synchronized and an incoherent phase at a critical value of the noise strength. Our results indicate that in the two phases as well as at the critical point, the autocorrelation for both the model decays as an exponential with a rate that increases continuously with the noise strength.