论文标题
二级库拉莫托模型在晶格上的同步过渡
Synchronization transition of the second-order Kuramoto model on lattices
论文作者
论文摘要
二阶Kuramoto方程描述了耦合振荡器与惯性的同步，例如，在功率电网中发生。与一阶库拉莫托方程相反，它的同步过渡行为鲜为人知。在高斯自以置信的情况下，这是不连续的，与一阶库拉马托方程的连续过渡相反。在这里，我们研究了对大型2D和3D晶格的过渡，并提供了混合相变的数值证据，即振荡器相$θ_i$显示出交叉，而频率则在3D中传播了实际相变。因此，预期频率的较低临界尺寸$ d_l^o = 2 $，而$ d_l^r = 4 $对于无数情况下的阶段。我们提供了关键指数的数值估计值，发现在与线性近似一致的相一致的情况下，频率扩展衰减为$ \ sim t^{ - d/2} $。但是，在3D中，在最初随机分布的$θ_i$的情况下，我们发现了一个更快的衰减，其特征是$ \ sim t^{ - 1.8（1）} $是由于随机相位波动出现的增强非线性的结果。
The second-order Kuramoto equation describes synchronization of coupled oscillators with inertia, which occur in power grids for example. Contrary to the first-order Kuramoto equation it's synchronization transition behavior is much less known. In case of Gaussian self-frequencies it is discontinuous, in contrast to the continuous transition for the first-order Kuramoto equation. Here we investigate this transition on large 2d and 3d lattices and provide numerical evidence of hybrid phase transitions, that the oscillator phases $θ_i$, exhibit a crossover, while the frequency spread a real phase transition in 3d. Thus a lower critical dimension $d_l^O=2$ is expected for the frequencies and $d_l^R=4$ for the phases like in the massless case. We provide numerical estimates for the critical exponents, finding that the frequency spread decays as $\sim t^{-d/2}$ in case of aligned initial state of the phases in agreement with the linear approximation. However in 3d, in the case of initially random distribution of $θ_i$, we find a faster decay, characterized by $\sim t^{-1.8(1)}$ as the consequence of enhanced nonlinearities which appear by the random phase fluctuations.
