| Project/Area Number |
15540376
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Mathematical physics/Fundamental condensed matter physics
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| Research Institution | Waseda University |
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Principal Investigator |
AIZAWA Yoji Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (70088855)
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| Project Period (FY) |
2003 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2005: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2003: ¥2,000,000 (Direct Cost: ¥2,000,000)
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| Keywords | 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 Theory / Darling-Kac-Aaronson theo / Large Deviation / Non-Stationarity / Levy Diffusion |
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| Research Abstract |
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.
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