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

作为具有灾难的复杂系统的一个例子，我们研究了互联网。我们进行了Ping实验，并分析了Ping信号的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着显著的共同特征。特别是，我们已经确定了古腾堡-Richetr法律和大森法律的互联网时间序列。接下来，我们着手研究在加州和日本取得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遵循Tsallis统计。我们还研究了与余震有关的大森区，发现存在一定的老化现象，它服从标度律。这一结果表明，余震的机制可以看作是一种玻璃动力学。在对地震的研究之后，我们又引入了地震网络的概念，它是一种新的地震网络。 ...更多信息 这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，发现其周期分布也是无标度的。除了这些唯象研究之外，我们还开展了非广延统计力学的基础理论研究。首先，我们已经证明了q-期望值形式主义与最小相对熵原理和最大Tsallis熵原理是一致的。其次，我们讨论了具有长程相互作用的非广延系统的热力学。我们证明了数值模型分析所建立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，我们考虑了爱因斯坦1905年的布朗运动理论中的反常扩散现象，并证明了分数福克-普朗克方程是如何通过放松爱因斯坦理论中二阶矩的存在而自然地导出的。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