Study of complex systems with catastrophes based on Tsallis' nonextensive statistical mechanics

作为具有灾变的复杂系统的一个例子，我们研究了互联网。进行了Ping实验，分析了Ping信号往返时间的时间序列数据。我们发现，互联网时间序列与地震时间序列有明显的共同特征。特别是，我们已经确定了互联网时间序列的古登堡-里希特定律和大森定律。接下来，我们继续调查加州和日本的地震时间序列数据。我们发现，两次连续地震之间的空间距离和时间间隔都服从Tsallis统计。我们还研究了与余震有关的大森体系，发现存在一定的老化现象，并服从标度律。这一结果表明，余震的机制可以被认为是一种玻璃动力学。在这些关于地震的研究之后，我们引入了地震网络的概念，Wh…更重要的是地震时间序列的演化随机网络映射。我们发现该网络是小世界类型的，并且是无标度的。此外，我们还研究了地震时间序列的有向网络特征，发现周期分布也是无标度的。除了这些唯象研究之外，我们还开展了非广泛统计力学的基础理论研究。首先，我们证明了q-期望值形式与与最大Tsallis熵原理相关的最小相对熵原理是一致的。其次，我们讨论了具有长程相互作用的非扩展系统的热力学。结果表明，数值模型分析所猜想的热力学标度关系可以在广义欧拉关系的基础上得到肯定的证明。第三，我们从爱因斯坦1905年的布朗运动理论出发，考虑了反常扩散现象，并证明了通过放宽爱因斯坦理论中二阶矩的存在，分数阶福克-普朗克方程是如何自然地得到的。较少

As an example of complex systems with catastrophes, we have studied the Internet. We have performed the Ping experiment and analyzed the time series data of the round-trip times of the Ping signals. We have discovered that there are striking common features of the Internet time series with seismic time series. In particular, we have identified the Gutenberg-Richetr law and the Omori law for the Internet time series. Next, we proceeded to investigate the seismic time series data taken in California and Japan. We have found that both the spatial distance and the time interval between two successive earthquakes follow Tsallis statistics. We have also studied the Omori regime relevant to aftershocks, and have discovered that there exist a definite aging phenomenon, which obeys a scaling law. This result suggests that the mechanism governing aftershocks can be thought of as a kind of glassy dynamics.After these studies on earthquakes, we have introduced the concept of earthquake network, wh … More ich is a evorvrng-random-network mapping of the seismic time series. We have found that the network is of the small-world type and is scale free. Also, we have investigated the directed-network feature of the seismic time series and seen that the period distribution is also scale free.In addition to these phenomenological studies, we have also developed fundamental theoretical studies on nonextensive statistical mechanics. Firstly, we have shown how the q-expectation value formalism is consistent with the minimum relative entropy principle associated with the maximum Tsallis entropy principle. Secondly, we have discussed thermodynamics ofa nonextensive system with a long-range interaction. We have shown that the thermodynamic scaling relation conjectured by numerical model analysis can affirmatively proved based on the generalized Euler relation. Thirdly, we have considered the phenomenon of anomalous diffusion in view of Einstein's 1905 theory of Brownian motion and have shown how naturally the fractional Fokker-Planck equation can be derived by relaxing the existence of the second moment in Einstein's theory. Less