Abe Sumiyoshi, Suzuki Norikazu, Tayurskii Dmitrii A
Department of Physics, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China.
Institute of Physics, Kazan Federal University, Kazan 420008, Russia.
Entropy (Basel). 2023 Jun 28;25(7):989. doi: 10.3390/e25070989.
The Omori-Utsu law shows the temporal power-law-like decrease of the frequency of earthquake aftershocks and, interestingly, is found in a variety of complex systems/phenomena exhibiting catastrophes. Now, it may be interpreted as a characteristic response of such systems to large events. Here, hierarchical dynamics with the fast and slow degrees of freedom is studied on the basis of the Fokker-Planck theory for the load-state distribution to formulate the law as a relaxation process, in which diffusion coefficient in the space of the load state is treated as a fluctuating slow variable. The evolution equation reduced from the full Fokker-Planck equation and its Green's function are analyzed for the subdynamics governing the load state as the fast degree of freedom. It is shown that the subsystem has the temporal translational invariance in the logarithmic time, not in the conventional time, and consequently the aging phenomenon appears.
大森-郁积定律表明地震余震频率随时间呈幂律式下降，有趣的是，在各种表现出灾难的复杂系统/现象中都能发现该定律。现在，它可以被解释为这类系统对大事件的一种特征响应。在此，基于福克-普朗克理论对负载状态分布进行研究，探讨具有快速和慢速自由度的层级动力学，将该定律表述为一个弛豫过程，其中负载状态空间中的扩散系数被视为一个波动的慢变量。针对作为快速自由度的负载状态的子动力学，分析了从完整福克-普朗克方程简化得到的演化方程及其格林函数。结果表明，子系统在对数时间而非传统时间上具有时间平移不变性，因此会出现老化现象。
