# Microscopic Cross-Correlations in the Finite-Size Kuramoto Model of Coupled Oscillators

## TL;DR

This paper investigates how finite-size effects in the Kuramoto model induce cross-correlations between oscillators, revealing a noise-like synchronization phenomenon in a deterministic setting.

## Contribution

It is the first to demonstrate that finite-size fluctuations in the Kuramoto model can cause deterministic cross-correlations similar to noise-induced synchronization.

## Key findings

- Finite-size fluctuations induce cross-correlations between oscillators.
- Cross-correlation strength depends on frequency difference and fluctuation intensity.
- Deterministic finite ensembles exhibit effects akin to stochastic noise.

## Abstract

Super-critical Kuramoto oscillators with distributed frequencies separate into two disjoint groups: an ordered one locked to the mean field, and a disordered one consisting of effectively decoupled oscillators -- at least so in the thermodynamic limit. In finite ensembles, in contrast, such clear separation fails: The mean field fluctuates due to finite-size effects and thereby induces order in the disordered group. To our best knowledge, this publication is the first to reveal such an effect, similar to noise-induced synchronization, in a purely deterministic system. We start by modeling the situation as a stationary mean field with additional white noise acting on a pair of unlocked Kuramoto oscillators. An analytical expression shows that the cross-correlation between the two increases with decreasing ratio of natural frequency difference and noise intensity. In a deterministic finite Kuramoto model, the strength of the mean field fluctuations is inextricably linked to the typical natural frequency difference. Therefore, we let a fluctuating mean field, generated by a finite ensemble of active oscillators, act on pairs of passive oscillators with a microscopic natural frequency difference between which we then measure the cross-correlation, at both super- and sub-critical coupling.