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

基于Tsallis非广延统计力学的复杂灾难系统研究

基本信息

批准号：
15540360
负责人：
ABE Sumiyoshi
金额：
$ 0.96万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

项目摘要

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

我们研究了互联网，作为具有灾难性的复杂系统的一个例子。我们进行了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显著的共同特征。特别是，我们已经确定了互联网时间序列的古腾堡-里切特定律和大森定律。接下来，我们对加利福尼亚和日本的地震时间序列数据进行了研究。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状态，发现存在一定的老化现象，并遵循标度规律。这一结果表明，控制余震的机制可以被认为是一种玻璃动力学。在对地震进行了这些研究之后，我们引入了地震台网的概念，它更丰富的是地震时间序列的演化-随机-网络映射。我们发现这个网络是小世界型的，是无标度的。此外，我们还研究了地震时间序列的定向网络特征，发现周期分布也是无标度的。除了这些现象学研究之外，我们还开展了非扩展统计力学的基础理论研究。首先，我们展示了q期望值的形式是如何与最小相对熵原理和最大Tsallis熵原理相一致的。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。在广义欧拉关系的基础上，证明了数值模型分析所推测的热力学标度关系是可以肯定地证明的。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第二矩的存在，分数式福克-普朗克方程是如何自然地推导出来的。少