To synchronize or not to synchronize, that is the question: finite-size scaling and fluctuation effects in the Kuramoto model

TL;DR

This paper investigates the finite-size scaling and fluctuation effects in the Kuramoto model, revealing how system size and fluctuations influence synchronization transitions in coupled oscillators.

Contribution

It provides new insights into the critical properties and finite-size scaling behavior of the Kuramoto model, including unconventional exponents and the role of fluctuations.

Findings

Finite-size effects significantly influence the synchronization transition.

Different types of frequency entrainment occur depending on the lattice dimension.

Unconventional critical exponents describe the transition behavior.

Abstract

The entrainment transition of coupled random frequency oscillators presents a long-standing problem in nonlinear physics. The onset of entrainment in populations of large but finite size exhibits strong sensitivity to fluctuations in the oscillator density at the synchronizing frequency. This is the source for the unusual values assumed by the correlation size exponent $\nu'$. Locally coupled oscillators on a $d$-dimensional lattice exhibit two types of frequency entrainment: symmetry-breaking at $d > 4$, and aggregation of compact synchronized domains in three and four dimensions. Various critical properties of the transition are well captured by finite-size scaling relations with simple yet unconventional exponent values.