Relaxation dynamics of the Kuramoto model with uniformly distributed natural frequencies

TL;DR

This paper investigates how the Kuramoto model with uniformly distributed natural frequencies relaxes near the phase transition, revealing that metastable states are finite-size effects that diminish as system size grows.

Contribution

It provides numerical evidence that metastability observed in finite systems is a finite-size effect, clarifying the relaxation dynamics near the phase transition.

Findings

Metastable states are finite-size effects that become rarer with larger systems.

Relaxation occurs as a step-like jump in the order parameter.

Metastability diminishes as system size increases.

Abstract

The Kuramoto model describes a system of globally coupled phase-only oscillators with distributed natural frequencies. The model in the steady state exhibits a phase transition as a function of the coupling strength, between a low-coupling incoherent phase in which the oscillators oscillate independently and a high-coupling synchronized phase. Here, we consider a uniform distribution for the natural frequencies, for which the phase transition is known to be of first order. We study how the system close to the phase transition in the supercritical regime relaxes in time to the steady state while starting from an initial incoherent state. In this case, numerical simulations of finite systems have demonstrated that the relaxation occurs as a step-like jump in the order parameter from the initial to the final steady state value, hinting at the existence of metastable states. We provide numerical evidence to suggest that the observed metastability is a finite-size effect, becoming an increasingly rare event with increasing system size.