Universal aspects of Malti-ergodic phase space structure in many body hamiltonian dynamics

多体哈密顿动力学中马尔蒂遍历相空间结构的普遍方面

基本信息

批准号：
15540376
负责人：
AIZAWA Yoji
金额：
$ 2.05万
依托单位：
Waseda University
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 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540376/
关键词：
Multi-ergodicity Non-hyperbolic System Infinite Ergodicity Lempel-Ziv Complexity SRB measure Weibull Distribution Nekoroshev Theorem Large Deviation Theory Multi-Ergodicity Log-Weibull Distribution Nekoroshev theorem Infinite Ergodic

项目摘要

The invariantt measure of generic hamiltonian systems reveals a number of singularities in phase space, where the non-hyperbolicity of hamiltonian dynamics plays an essential role. The appearance of such infinite measure has been studied so far in the framework of infinite ergodic theory [Aaronson, 1997]. [A] We have studied many characteristics of non-hyperbolic chaos, such as the spectral indeces, correlation functions, Lyapunov exponents, and the Lempel-Ziv Complexity, and we have succeeded to derive the exact interrelations among those characteristics. Furthermore, we have succeeded to formulate the exact Levy diffusion equation in terms of anomalous diffusion in the phase space of non-hyperbolic hamiltonian dynamics. These results suggest that the measure-theoretical structure of hamilton dynamics is quite different from the ordinary SRB measures in hyperbolic systems. [B] The infinite ergodic aspects are discovered in the cluster formation process, where the Weibull type distributions are universally observed. We have succeeded to explain those statistical laws from the scaling theory based on the Nekoroshev theorem, and also revealed that the large deviation properties are realized in those statistical features. These results suggest very strongly that the Arnold diffusion may be characterized in the framework of the infinite ergodic theory.

通用汉密尔顿系统的不变性量度揭示了相位空间中的许多奇异性，其中汉密尔顿动力学的非透明度起着至关重要的作用。到目前为止，在无限的ergodic理论的框架中已经研究了这种无限度量的出现[Aaronson，1997]。 [a]我们研究了非纤维混乱的许多特征，例如光谱固定，相关函数，lyapunov指数和lempel-ziv复杂性，并且我们成功地得出了这些特征之间的确切相互关系。此外，我们已经成功地根据非氧化汉密尔顿动力学的相空间中的异常扩散来制定确切的征税扩散方程。这些结果表明，汉密尔顿动力学的测量理论结构与双曲系统中普通的SRB测量截然不同。 [b]在集群形成过程中发现了无限的厄运方面，在该过程中，在其中普遍观察到了威布尔类型分布。我们已经成功地基于Nekoroshev定理来解释从缩放理论中的统计定律，并揭示了在这些统计特征中实现了较大的偏差特性。这些结果非常强烈地表明，在无限的ergodic理论的框架中，Arnold扩散可能是特征的。