Finite-size scaling, dynamic fluctuations, and hyperscaling relation in the Kuramoto model

TL;DR

This paper investigates how finite-size effects and frequency sampling methods influence the synchronization transition in the Kuramoto model, revealing distinct scaling behaviors and the validity of hyperscaling depending on the sampling approach.

Contribution

It demonstrates that frequency sampling methods significantly affect finite-size scaling and hyperscaling relations in the Kuramoto model, providing new insights into synchronization phenomena.

Findings

Different sampling methods lead to distinct finite-size scaling exponents.

Hyperscaling relation holds in regular sampling but is violated in random sampling.

Mean-field theory predicts the FSS exponent but not the critical amplitude in the random case.

Abstract

We revisit the Kuramoto model to explore the finite-size scaling (FSS) of the order parameter and its dynamic fluctuations near the onset of the synchronization transition, paying particular attention to effects induced by the randomness of the intrinsic frequencies of oscillators. For a population of size $N$, we study two ways of sampling the intrinsic frequencies according to the {\it same} given unimodal distribution $g(\omega)$. In the `{\em random}' case, frequencies are generated independently in accordance with $g(\omega)$, which gives rise to oscillator number fluctuation within any given frequency interval. In the `{\em regular}' case, the $N$ frequencies are generated in a deterministic manner that minimizes the oscillator number fluctuations, leading to quasi-uniformly spaced frequencies in the population. We find that the two samplings yield substantially different finite-size properties with clearly distinct scaling exponents. Moreover, the hyperscaling relation between the order parameter and its fluctuations is valid in the regular case, but is violated in the random case. In this last case, a self-consistent mean-field theory that completely ignores dynamic fluctuations correctly predicts the FSS exponent of the order parameter but not its critical amplitude.