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

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.

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• Origin and scaling of chaos in weakly coupled phase oscillators
• Critical behavior of noise-induced phase synchronization
• Strange Scaling and Temporal Evolution of Finite-Size Fluctuation in Thermal Equilibrium
• Twisted states in low-dimensional hypercubic lattices
• Microscopic Cross-Correlations in the Finite-Size Kuramoto Model of Coupled Oscillators
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• Instantaneous frequencies in the Kuramoto model
• Universality of critical dynamics on a complex network
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• Predicting the Oscillatory Regimes of Global Synchrony Induced by Secondary Clusters
• Data-driven stochastic modeling of coarse-grained dynamics with finite-size effects using Langevin regression
• Efficient moment-based approach to the simulation of infinitely many heterogeneous phase oscillators
• Nondivergent and negative susceptibilities around critical points of a long-range Hamiltonian system with two order parameters