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Numerical Analysis of Generalized Statistical Manifolds Associated with Stochastic Processes

Research Project

Project/Area Number10680322
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Statistical science
Research InstitutionGunma National College of Technology

Principal Investigator

OBATA Tuhiriro  Gunma National College of Technology, Department of Electrical Engineering, Professor, 電気工学科, 教授 (50005534)

Co-Investigator(Kenkyū-buntansha) OSHIRAA HiroshiToho  Toho University, School of Medicine, lecturer, 医学部, 講師 (30104152)
HARA Hiroaki  Tohoku University, Graduate School of Engineering, Professor, 大学院・工学研究科, 教授 (60005296)
Project Period (FY) 1998 – 2000
Project Status Completed(Fiscal Year 2001)
Budget Amount *help
¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 2000 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Keywordsstatistical manifold / information geometry / stochastic process / curvature / Higgs field / quantum gases / Gentitle statistics / uncertainty relation / 双対構造 / 中間統計 / 分数統計 / ゆらぎ / ペアロン / ランダムウォーク / 歩行運動 / 1 / fゆらぎ / 曲率テンソル / Newton-Cartan理論 / ゲージ場 / Riemann-Liouville積分
Research Abstract

1.A generalized statistical manifold is constructed for (time-) discrete stochastic processes characterized by n parameters 6=(θ_1・・・θ_n). According to information geometry, a metric tensor and an α-connection are introduced on an n-parameter family S_N={p(X, θ, N) θ∈Ω} of probability density functions p(X, θ, N) for a random variable X(N) at a discrete time N. As the discrete time goesby, a time series of statistical manifolds・・・, S_1, S_2, ・・・, S_N・・・ is generated, andgathering them leads to a stratified statistical manifold, which is the direct product space S×Z of a statistical manifold S and a set Z of integers. On the generalized statistical manifold an extended connection connecting two layers is introduced after the Newton-Cartan theory of Newton gravity and a theory in elementary particle physics in which Higgs fields are formulated as extended gauge fields. And then extended curvature tensors are constructed from the extended connection and the α-connection. Applying this new geometrical method to a random walk model, we find that its generalized statistical manifold has a duality structure.
2.The physical role of the Riemann scalar curvature R of thermodynamic equilibrium systems is studied. Janyszek and Mrugala (JM) interpreted the tfas a measure of the instability of the systems.
3.To examine wider roles of curvatures in physics, we also construct new models of thermodynamic systems or stochastic processes. Three germs are obtained, but geometrical characteristics in these models are left as future works.

Report

(4results)
  • 2001 Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • 1998 Annual Research Report

Research Products

(31results)

All Other

All Publications

  • [Publications] Tsunehiro OBATA: "Differential Geometry of Discrete Stochastic Processes"Journal of the Korean Physical Society. 32. 773-785 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 原 啓明: "複雑な粘弾性物質のモデル化と逆問題:Riemann-Liouville積分表示"統計数理. 46. 477-491 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Oshima: "Riemann scalar curvature of ideal quantum gases obeying Gentile's statistics"J.Phys.A : Math.Gen.. 32. 6373-6383 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] SHIGEJI FUJITA: "ON THE SUSCEPTIBILITY IN La_<2-X>Sr_XCuO_4"Modern Physics Letters B. 14. 849-858 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tsunehiro Obata: "Duality in Generalized Statistical Manifolds"Journal of the Korean Physical Society. 38. 475-479 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Oshima: "Geometrical Expression of Thermal Fluctuations"Journal of the Korean Physical Society. 38. 486-489 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shigeji Fujita: "Theory of the Magnetic Susceptibility in La_<2-X>Sr_XCuO_4"Physical Review B. 63. 054402(6) (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Hara: "The 5^<th> International Workshop on SIMILARITY IN DIVERSITY"H.Hara. 198 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Tokuyama: "STATISTICAL PHYSICS"AIP. 800 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] The Electromagnetics Academy: "PIERS 2000 Proceedings"The Electromagnetics Academy. 1178 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tsunehiro OBATA: "Differential Geometry of Discrete Stochast Processes"Journal of the Korean Physical Society. 32. 773-785 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroaki Kara: "Modelling of Complicat'ed Visco-elastic Material and Its Inverse Problem : Riemann-Liouville Integral Representation"Proceedings of the Institute of Statistical Mathematics. 46. 477-491 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Oshima: "Riemann scalar curvature of ideal quantum gases obeying Gentile' s statistics"J. Phys. A : Math. Gen. 32. 6373-6383 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] SHIGEJI FUJITA: "ON THE SUSCEPTIBILITY IN La_<2a-x> Sr_x CuO_4"Modern Physics Letters B. 14. 849-858 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tsunehiro Obata: "Duality in Generalized Statistical Manifolds"Journal of the Korean Physical Society. 38. 475-479 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H, Oshima: "Geometrical Expression of Thermal Fluctuations"Journal of the Korean Physical Soc. 38. 486-489 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Hara: "The 5th International Workshop on SIMILARITY IN DIVERSITY"H. Hara. 198 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Tokuyama: "STATISTICAL PHYSICS"AIP. 800 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] The Electromagnetics Academy: "PIERS 2000 Proceedings"The Electromagnetics Academy. 1178 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Obata: "Duality in Generalized Statisitcal Manifolds"Journal of Korean Physical Society. 38. (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Oshima: "Geometrical expression of the thermal fluctuations"Journal of Korean Physical Society. 38. (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Fujita: "Theory of the Magnetic Susceptibility in La_<2-X>Sr_XCuO_4"Phys.Rev.B. 63. 054402(6) (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Fujita: "On the susceptibility in La_<2-X>Sr_XCuO_4"Modern Physics Letters B. 14・24. 849-858 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Obata: "Approach to Discrete Stochastic Processes from the Viewpoint of Newton-Cartan Theory of Gravity"PIERS 2000 Proceedings. 658 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Hiroshi Oshima: "Riemann scalar curvature of ideal quantum gases obeying Gentile's statistics"J. Phys. A : Math. Gen. 32. 6373-6383 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Oshima: "Scalar Curvature of Massive Ideal Gases in Two Dimensions"The 5th Workshop on SIMILARITY IN DIVERSITY. 153-156 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Obata: "Fluctuations in human's walking"The 3rd Tohwa University International Conference on Statistical Physics. 145-145 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Oshima: "Scalar curvature of massive ideal gases"The 3rd Tohwa University International Conference on Statistical Physics. 149-149 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Tsunehiro OBATA: "Differential Geometry of Discrete Stochatic Processes" Journal of the Korean Physical Society. 32・6. 773-785 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 原 啓明: "複雑な粘弾性物質のモデル化と逆問題:Riemann-Liouviloe積分表示" 統計数理. 46・2. 477-491 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Tokuyama: "STATISTICAL PHYSICS" World Scientific, 244 (1998)

    • Related Report
      1998 Annual Research Report

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Published : 1999-04-01   Modified : 2016-04-21  

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