| Project/Area Number |
09640472
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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 |
物性一般(含基礎論)
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| Research Institution | Waseda University |
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Principal Investigator |
AIZAWA Yoji Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (70088855)
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| Co-Investigator(Kenkyū-buntansha) |
MIYASAKA Tomohiro ATR Adaptive Communication Research Laboratorie, Researcher, 第四研究室, 研究員 (90257246)
HARAYAMA Takahisa ATR Adaptive Communication Research Laboratories, Researcher, 第四研究室, 研究員 (70247229)
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| Project Period (FY) |
1997 – 2000
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| Project Status |
Completed (Fiscal Year 2001)
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| Budget Amount *help |
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2000: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1999: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 1998: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1997: ¥1,600,000 (Direct Cost: ¥1,600,000)
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| Keywords | Hamiltonian Systems / Chaos / KAM / Invariant Tori / Cluster / Weibull Distributions / Quantum Chaos / Energy Level Statistics / 近可積分系 / クラスター / ベリー・ロブニック公式 / 三体問題 / KAMトーラス / ノン・ツイスト系 / ア-ノルド拡散 / 分岐現象 / ノン・ツイスト写像 |
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| Research Abstract |
In order to elucidate the universal laws of Hamiltonian dynamical systems, we studied ergodic and kinetic properties of trajectories generated by complex phase space structures. For various Hamiltonian systems, phase space phenomena such as breakup of invariant tori and bifurcations of periodic trajectories were thoroughly studied with particular interest in their effects on the long-term behavior of trajectories for classical systems and on the energy level statistics for quantum systems. In particular, for many particle systems, we developed a kinetic theory of complex phenomena such as clustering motions, relaxation processes and anomoulous diffusion. In addition, we pursued the possibility to apply our theoretical results to real experiments in the optical laser systems, where Hamiltonian dynamical theory can be applicable.
Main results are summarized as follows :
(1) Fractal structures and phase transition phenomena in mixed Hamiltonian systems
Universal scaling laws are revealed in the transition regime between tori and chaos. It is shown that many particle systems exhibit statistical distributions such as Weibull distribution and Log-Weibull distribution, which are consistent with the Arnold diffusion theory.
(2)Universal properties of Hamiltonian systems violating the KAM condition
A new method is proposed to systematically investigate the phase space of Hamiltonian systems violating the KAM condition. Using this method, we succeed in analyzing the reconnection phenomena and accurately determining the critical threshold for global chaos. It is shown that the set of critical threshold constitutes a fractal.
3) Quantum signatures of classical phase space phenomena
We establish a systematic method to determine the Berry-Robnik parameter of energy level statistics for non-integrable billiard systems. On the basis of this method, we demonstrate that the classical bifurcation phenomena clearly effect the level statistics properties of the corresponding quantum systems.
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