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.

通用哈密顿系统的不变测度揭示了相空间中的许多奇点，其中哈密顿动力学的非双曲性起着至关重要的作用。迄今为止，这种无限测度的出现已经在无限遍历理论的框架中进行了研究[Aaronson，1997]。 [A] 我们研究了非双曲混沌的许多特征，例如谱因数、相关函数、Lyapunov 指数和 Lempel-Ziv 复杂度，并且我们成功地推导了这些特征之间的精确相互关系。此外，我们成功地根据非双曲哈密顿动力学相空间中的反常扩散制定了精确的 Levy 扩散方程。这些结果表明，汉密尔顿动力学的测度理论结构与双曲系统中普通的 SRB 测度有很大不同。 [B] 在簇形成过程中发现了无限遍历方面，普遍观察到威布尔型分布。我们成功地从基于涅科罗舍夫定理的标度理论中解释了这些统计规律，并揭示了这些统计特征中实现了大偏差特性。这些结果强烈表明阿诺德扩散可以在无限遍历理论的框架中进行表征。