The Kuramoto model, which serves as a paradigm for investigating synchronization phenomena of oscillatory systems, is known to exhibit second-order, i.e., continuous, phase transitions in the macroscopic order parameter. Here we generalize a number of classical results by presenting a general framework for analytically capturing the critical scaling properties of the order parameter at the onset of synchronization. Using a self-consistent approach and constructing a characteristic function, we identify various phase transitions toward synchrony and establish scaling relations describing the asymptotic dependence of the order parameter on the coupling strength near the critical point. We find that the geometric properties of the characteristic function, which depends on the natural frequency distribution, determine the scaling properties of order parameter above criticality.

中文翻译：

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耦合相位振荡器群的通用缩放和相变

已知作为研究振荡系统的同步现象的范例的仓本模型在宏观阶参数中表现出二阶即连续的相变。在这里，我们通过提出一个通用框架来概括许多经典结果，该框架用于在同步开始时分析性捕获顺序参数的关键缩放属性。使用自洽方法并构造特征函数，我们确定了朝着同步的各种相变，并建立了描述临界点附近临界点上耦合参数的渐进依赖关系的比例关系。我们发现特征函数的几何特性取决于自然频率分布，