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2006 Fiscal Year Final Research Report Summary

Geometric Structure of Multi-dimensional Chaos and Its Application to Reactions in Non-equilibrium

Research Project

Project/Area Number 16340113
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Mathematical physics/Fundamental condensed matter physics
Research InstitutionNara Women's University

Principal Investigator

TODA Mikito  Nara Women's University, Faculty of Science, Associate Professor (70197896)

Co-Investigator(Kenkyū-buntansha) KOMATSUAZKI Tamiki  Kobe University, Faculty of Science, Associate Professor (30270549)
Project Period (FY) 2004 – 2006
KeywordsDynamical Systems / Chaos / Statistical Physics / Chemical Reactions / Non-equilibrium / Geometric Structures / Complex Systems / Invariant manifolds
Research Abstract

We have formulated the dynamical theory of reactions based on the concept of normally hyperbolic invariant manifolds (NHIMs). This enables us to obtain a mathematically firm foundation of the concept of "transition states" in reaction processes. Moreover, it offers a possibility of going over the conventional ideas of reactions. The first is the possibility of breakdown of the normal hyperbolicity caused by chaos on the NHIM. Note that normal hyperbilicity means that instability along the normal directions is larger than that along the tangential directions. However, when chaos on the NHIM becomes comparable to the instability along the normal directions, the normal hyperbolicity is at the edge of breaking down. We have constructed a model system where this phenomenon can be analyzed using the Lie transformation and Pade summation. The breakdown means, in the context of reactions, that a new set of degrees of freedom start to contribute collective degrees of freedom describing reactions. This opens a new horizon in studying the reaction processes. The second achievement is to indicate that a power-law behavior is observed when the dynamics in the potential well is non-ergodic. We have constructed a model where the Arnold web in the well is sparse and non-uniform. The distribution of residence times exhibit a power-law properties and the dynamics in the well show anomalous diffusion. These phenomena indicate that the traditional concept of the reaction rate constat does not exists any more. The above two aspects of the reaction processes indicate that the dynamical theory of reactions offer a new possibility of understanding reactions.

  • Research Products

    (15 results)

All 2008 2007 2006

All Journal Article (15 results)

  • [Journal Article] Foundation and Limitations of Statistical Reaction Theory2008

    • Author(s)
      Shojiguchi, Li, Komatsuzaki, Toda
    • Journal Title

      Communications in Nonlinear Science and Numerical Simulation 13

      Pages: 857-867

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Foundation and Limitations of Statistical Reaction Theory2008

    • Author(s)
      Shoujiguchi, Li, Komatsuzaki, Toda
    • Journal Title

      Communications in Nonlinear Science and Numerical Simulation 13

      Pages: 857-867

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Fractional Behavior in Nonergodic reaction Processes of Isomerization2007

    • Author(s)
      Shoujiguchi, Li, Komatsuzaki, Toda
    • Journal Title

      Physical Review E 75

      Pages: 035204(4 )

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Dimension Reduction for Extracting Geometrical Structure of Multidimensional Phase Space: Application to Fast Energy Exchange in the Reaction O(1D)+N2O→NO+NO2007

    • Author(s)
      Kawai, Fujimura, Kajimoto, Yamashita, Li, Komatsuzaki, Toda
    • Journal Title

      Physical Review A 75

      Pages: 022714(11 )

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Fractional Behavior in Multi-dimensional Hamiltonian Systems Describing Reactions2007

    • Author(s)
      Shojiguchi, Li, Komatsuzaki, Toda
    • Journal Title

      Phys. Rev. E 75

      Pages: 022714(11 )

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Fractional Behavior in Nonergodic reaction Processes of Isomerization2007

    • Author(s)
      Shouj iguchi, Li, Komatsuzaki, Toda
    • Journal Title

      Phys. Rev. E 75

      Pages: 035204(R), 4

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Dimension Reduction for Extracting Geometrical Structure of Multidimensional Phase Space : Application to Fast Energy Exchange in the Reaction 0(1D)+N20-> NO+NO2007

    • Author(s)
      Kawai, Fujimura, Kajimoto, Yamashi ta, Li, Komatsuzaki, Toda
    • Journal Title

      Phys. Rev. A 75

      Pages: 022714, 11

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Fractional Behavior in Multi-dimensional Hamiltonian Systems Describing Reactions2007

    • Author(s)
      Shoujiguchi, Li, Komatsuzaki, Toda
    • Journal Title

      Phys. Rev. E 75

      Pages: 022714, 11

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Universality in Dynamical Formation of Entanglement in Quantum Chaos2006

    • Author(s)
      Kubotani, Toda, Adachi
    • Journal Title

      Physical Review A 74

      Pages: 032314(5 )

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Definability of no-return transition states in high energy regime above reaction threshold2006

    • Author(s)
      Li, Shojiguchi, Toda, Komatsuzaki
    • Journal Title

      Physical Review Letters 97

      Pages: 028302(4 )

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Dynamical Hierarchy in Transition States of Reactions2006

    • Author(s)
      Li, Shojiguchi, Toda, Komatsuzaki
    • Journal Title

      Few Body Systems 38

      Pages: 173-179

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Wavelet Analysis and Arnold Web Picture for Detecting Energy Transfer in Hamiltonian Dynamical System2006

    • Author(s)
      Shojiguchi, Baba, Li, Komatsuzaki, Toda
    • Journal Title

      Laser Physics 16

      Pages: 1097-1106

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Universality in Dynamical Formation of Entanglement in Quantum Chaos2006

    • Author(s)
      Kubotani, Toda, Adachi
    • Journal Title

      Phys. Rev. A74

      Pages: 032314, 5

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Definability of no-return transition states in high energy regime above reaction threshold2006

    • Author(s)
      Li, Shojiguchi, Toda, Komatsuzaki
    • Journal Title

      Phys. Rev. Lett. 97

      Pages: 028302, 4

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Wavelet Analysis and Arnold Web Picture for Detecting Energy Transfer in Hamiltonian Dynamical System2006

    • Author(s)
      Shoujiguchi, Baba, Li, Komatsuzaki, Toda
    • Journal Title

      Laser Physics 16

      Pages: 1097-1106

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2010-02-04  

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