Ódor Géza, Deng Shengfeng
Centre for Energy Research, Institute of Technical Physics and Materials Science, P.O. Box 49, H-1525 Budapest, Hungary.
Entropy (Basel). 2023 Jan 13;25(1):164. doi: 10.3390/e25010164.
The second-order Kuramoto equation describes the synchronization of coupled oscillators with inertia, which occur, for example, in power grids. On the contrary to the first-order Kuramoto equation, its synchronization transition behavior is significantly less known. In the case of Gaussian self-frequencies, it is discontinuous, in contrast to the continuous transition for the first-order Kuramoto equation. Herein, we investigate this transition on large 2D and 3D lattices and provide numerical evidence of hybrid phase transitions, whereby the oscillator phases θi exhibit a crossover, while the frequency is spread over a real phase transition in 3D. Thus, a lower critical dimension dlO=2 is expected for the frequencies and dlR=4 for phases such as that in the massless case. We provide numerical estimates for the critical exponents, finding that the frequency spread decays as ∼t-d/2 in the case of an aligned initial state of the phases in agreement with the linear approximation. In 3D, however, in the case of the initially random distribution of θi, we find a faster decay, characterized by ∼t-1.8(1) as the consequence of enhanced nonlinearities which appear by the random phase fluctuations.
二阶Kuramoto方程描述了具有惯性的耦合振子的同步现象，例如在电网中就会出现这种情况。与一阶Kuramoto方程相反，其二阶同步转变行为鲜为人知。在高斯自频率的情况下，它是不连续的，这与一阶Kuramoto方程的连续转变形成对比。在此，我们研究了二维和三维大晶格上的这种转变，并提供了混合相变的数值证据，即振子相位θi呈现出交叉现象，而频率在三维空间中经历真正的相变。因此，对于频率，预期较低的临界维度dlO = 2，对于相位，预期较低的临界维度dlR = 4，就像无质量情况下那样。我们给出了临界指数的数值估计，发现在相位初始排列状态下，频率展宽按 ∼t-d/2衰减，这与线性近似一致。然而，在三维空间中，当初始θi随机分布时，我们发现衰减更快，其特征为 ∼t-1.8(1)，这是由随机相位涨落引起的增强非线性导致的。
