Slow spectral dynamics of shot noise in the Kuramoto model: the role of microscopic regularity

TL;DR

This study reveals that the microscopic arrangement of natural frequencies in the Kuramoto model significantly influences collective fluctuations, leading to slow spectral dynamics not predicted by random sampling assumptions.

Contribution

It demonstrates that deterministic frequency arrangements cause anomalously slow oscillations in the spectral dynamics, contrasting with random sampling results.

Findings

Deterministic frequency selection results in wave-like spectral patterns.

Oscillation periods scale linearly with system size.

Resonant interactions due to regular frequency structure drive slow dynamics.

Abstract

Finite-size effects in the Kuramoto model are known to induce collective fluctuations even below the critical coupling, where the thermodynamic limit predicts complete asynchrony. While the shot-noise approach developed in our recent work accurately describes the power spectrum of these fluctuations for random frequency sampling, the present study reveals that the microscopic realization of the frequency distribution plays a crucial role. We show that a deterministic (quasi-uniform) selection of natural frequencies from the same Lorentzian distribution leads to qualitatively different dynamics: the shot noise spectrum exhibits anomalously slow oscillatory behavior, manifesting as wave-like patterns in time-frequency representations. The period of these oscillations scales linearly with the system size and matches the frequency spacing between neighboring oscillators near the distribution center. Numerical simulations confirm that these slow spectral dynamics arise from resonant interactions facilitated by the regular frequency structure, which are absent for random sampling. Our findings demonstrate that identical integral frequency distributions do not guarantee equivalent collective dynamics, highlighting the necessity of accounting for the fine structure of microscopic parameters in finite-size populations.