Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.
Chaos. 2012 Mar;22(1):013133. doi: 10.1063/1.3692966.
We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models such as the Ising model, the scaling property of D is D∼O(1) for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of D is different for the coherent and incoherent regimes: D∼O(1/N(a)) with a certain constant a>0 in the coherent regime and D∼O(1) in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.
我们研究了全局耦合非同相同步相振荡器中 Kuramoto 序参量时间积分的扩散系数。该系数表示序参量时间积分与其平均值在样本平均上的偏差。换句话说，该系数表征了序参量的长期波动。对于 N 个耦合振荡器的系统，我们引入了一个统计量 D，表示 N 和扩散系数的乘积。我们研究了 D 相对于系统大小 N 的标度律。在其他著名模型中，如伊辛模型，除了相变点之外，D 的标度性质对于相干和非相干两种情况都是 D∼O(1)。相比之下，在全局耦合相振荡器中，D 的标度律在相干和非相干两种情况下是不同的：在相干情况下，D∼O(1/N(a))，其中 a 是某个大于零的常数，而在非相干情况下，D∼O(1)。我们证明了这些标度律对于几种有代表性的耦合方案都成立。
