中国物理B ›› 2025, Vol. 34 ›› Issue (10): 100501-100501.doi: 10.1088/1674-1056/addd84

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Condensation and criticality of eigen microstates of phase fluctuations in Kuramoto model

Ning-Ning Wang(王宁宁)1,2, Qing Yao(姚卿)3, Ying Fan(樊瑛)3, Zeng-Ru Di(狄增如)3,4, and Xiao-Song Chen(陈晓松)3,5,†   

  1. 1 School of Artificial Intelligence, Tiangong University, Tianjin 300387, China;
    2 School of Mathematical Science, Nankai University, Tianjin 300071, China;
    3 School of Systems Science, Beijing Normal University, Beijing 100875, China;
    4 International Academic Center of Complex Systems, Beijing Normal University, Zhuhai 519087, China;
    5 Institute for Advanced Study in Physics and School of Physics, Zhejiang University, Hangzhou 310058, China
  • 收稿日期:2025-04-27 修回日期:2025-05-25 接受日期:2025-05-28 发布日期:2025-10-09
  • 通讯作者: Xiao-Song Chen E-mail:chenxs@bnu.edu.cn
  • 基金资助:
    The authors would like to thank Dr Teng Liu, Dr Xiaojie Chen, Dr Gaoke Hu and Dr Jiaqi Dong for the insightful and valuable discussions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 12135003, 71731002, and 12471141), the Postdoctoral Fellowship Program of CPSF (Grant No. GZC20231179), the China Postdoctoral Science Foundation–Tianjin Joint Support Program (Grant No. 2023T001TJ), and the Tianjin Education Commission scientific Research Project (Grant No. 2023SK070).

Condensation and criticality of eigen microstates of phase fluctuations in Kuramoto model

Ning-Ning Wang(王宁宁)1,2, Qing Yao(姚卿)3, Ying Fan(樊瑛)3, Zeng-Ru Di(狄增如)3,4, and Xiao-Song Chen(陈晓松)3,5,†   

  1. 1 School of Artificial Intelligence, Tiangong University, Tianjin 300387, China;
    2 School of Mathematical Science, Nankai University, Tianjin 300071, China;
    3 School of Systems Science, Beijing Normal University, Beijing 100875, China;
    4 International Academic Center of Complex Systems, Beijing Normal University, Zhuhai 519087, China;
    5 Institute for Advanced Study in Physics and School of Physics, Zhejiang University, Hangzhou 310058, China
  • Received:2025-04-27 Revised:2025-05-25 Accepted:2025-05-28 Published:2025-10-09
  • Contact: Xiao-Song Chen E-mail:chenxs@bnu.edu.cn
  • Supported by:
    The authors would like to thank Dr Teng Liu, Dr Xiaojie Chen, Dr Gaoke Hu and Dr Jiaqi Dong for the insightful and valuable discussions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 12135003, 71731002, and 12471141), the Postdoctoral Fellowship Program of CPSF (Grant No. GZC20231179), the China Postdoctoral Science Foundation–Tianjin Joint Support Program (Grant No. 2023T001TJ), and the Tianjin Education Commission scientific Research Project (Grant No. 2023SK070).

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摘要： The Kuramoto model is one of the most profound and classical models of coupled phase oscillators. Because of the global couplings between oscillators, its precise critical exponents can be obtained using the mean-field approximation (MFA), where the time average of the modulus of the mean-field is defined as the order parameter. Here, we further study the phase fluctuations of oscillators from the mean-field using the eigen microstate theory (EMT), which was recently developed. The synchronization of phase fluctuations is identified by the condensation and criticality of eigen microstates with finite eigenvalues, which follow the finite-size scaling with the same critical exponents as those of the MFA in the critical regime. Then, we obtain the complete critical behaviors of phase oscillators in the Kuramoto model. We anticipate that the critical behaviors of general phase oscillators can be investigated by using the EMT and different critical exponents from those of the MFA will be obtained.

关键词: synchronization, coupled oscillators, eigen microstate, criticality

Abstract: The Kuramoto model is one of the most profound and classical models of coupled phase oscillators. Because of the global couplings between oscillators, its precise critical exponents can be obtained using the mean-field approximation (MFA), where the time average of the modulus of the mean-field is defined as the order parameter. Here, we further study the phase fluctuations of oscillators from the mean-field using the eigen microstate theory (EMT), which was recently developed. The synchronization of phase fluctuations is identified by the condensation and criticality of eigen microstates with finite eigenvalues, which follow the finite-size scaling with the same critical exponents as those of the MFA in the critical regime. Then, we obtain the complete critical behaviors of phase oscillators in the Kuramoto model. We anticipate that the critical behaviors of general phase oscillators can be investigated by using the EMT and different critical exponents from those of the MFA will be obtained.

Key words: synchronization, coupled oscillators, eigen microstate, criticality

中图分类号:  (Synchronization; coupled oscillators)

  • 05.45.Xt
05.70.Fh (Phase transitions: general studies) 64.60.Ht (Dynamic critical phenomena)