2011 Fiscal Year Final Research Report
Multi ergodicity in nearly integrable Hamiltonian systems and large deviation properties of infinite ergodic systems
| Project/Area Number |
21540399
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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 早稲田大学, 理工学術院, 教授 (70088855)
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| Co-Investigator(Renkei-kenkyūsha) |
SHINKAI Soya 神戸大学, 大学院・システム情報学研究科, 学術研究員 (60547058)
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| Project Period (FY) |
2009 – 2011
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| Keywords | 非平衡 / 非線形物理学 |
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| Research Abstract |
In the present works, we have challenged to derive the scaling law of the orbital diffusion(Arnold diffusion) in high dimensional Hamiltonian systems, and succeeded to obtain the log-diffusion as well as the sub-diffusion in the interface layer between the invariant torus and chaos in a typical Hamiltonian system(Froeschle's model). Furthermore, it was proved these results can be theoretically formulated by use of the infinite ergodic theorem(DKA theorem), which is applicable to the Arnold webs in the nearby integrable Hamiltonian cases. From this theoretical result, we have succeeded to understand systematically the various aspects of slow dynamics in Hamiltonian systems, for instance, 1/f spectral fluctuations and the log-Weibull distribution function for the Nekholoshev time, which are closely connected to a universal distribution function(the Mittag-Leffler distribution) for the large deviation property of scaled average fluctuations. We have also succeeded to extend the DKA theorem to elucidate the detailed measure theoretical structure hidden behind the infinite ergodic phenomena in Hamiltonian systems, for instance, the fine structure of phase space(Mushroom billiard system) with a fractal scaling and the apparent attenuation in the transient regime between torus and chaos in nearby integrable systems. All these results are independent and new ones that stimulate the future development of the ergodic theoretical studies not only in Hamiltonian dynamics but also in the dissipative dynamics. Indeed, Strange-Nonchaotic Attractors and Heteroclinic turbulence reveal the same type of slow dynamics as these mentioned in our present works.
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