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2019 年度 実績報告書

準ハミルトン系の統計力学:非可積分トポロジー束縛による構造形成と非楕円型拡散過程

研究課題

研究課題/領域番号 18J01729
研究機関京都大学

研究代表者

佐藤 直木  京都大学, 数理解析研究所, 特別研究員(PD)

研究期間 (年度) 2018-04-25 – 2021-03-31
キーワードMetriplectic brackets / Euler Equations / Stellarator / Degenerate-Elliptic PDE / Elliptic-Hyperbolic PDE / Beltrami fields
研究実績の概要

The aim of the present research is to formulate the statistical theory of mechanical systems subject to non-integrable topological constraints, and to create the mathematical objects, concepts, and methods that are required to achieve this goal. The following results were obtained:
1、Ideal systems exhibit a Poisson structure. However, the algebraic structure of non-ideal systems is an open issue. Here, we showed that the Fokker-Planck equation describing diffusion processes in noncanonical Hamiltonian systems exhibits a metriplectic structure, i.e. an algebraic formalism that generates the equation in consistency with the thermodynamic principles of energy conservation and entropy growth.
2、The statistical properties of topologically constrained mechanical systems can be related to the geometric properties of stationary solutions of the ideal Euler equations. Here, we investigated the existence of stationary solutions of the Euler equations without continuous Euclidean symmetries and with non-vanishing pressure gradients, and provided smooth analytic examples in bounded domains.
3、The Sobolev-like Hilbert space of the solutions of the second order degenerate-elliptic partial differential equation (orthogonal Poisson equation) object of this study and the associated topology were examined. These results were reported in N Sato and Z Yoshida 2019 J. Phys. A: Math. Theor. 52 355202. Here, it is found that a non-vanishing helicity compensates the broken ellipticity.
4、The existence of Beltrami operators in dimensions higher than 3 was shown. Analytic examples were given.

現在までの達成度 (段落)

翌年度、交付申請を辞退するため、記入しない。

今後の研究の推進方策

翌年度、交付申請を辞退するため、記入しない。

  • 研究成果

    (14件)

すべて 2019 その他

すべて 雑誌論文 (5件) (うち査読あり 5件) 学会発表 (6件) (うち国際学会 3件、 招待講演 3件) 備考 (3件)

  • [雑誌論文] Existence of ideal magnetofluid equilibria without continuous Euclidean symmetries2019

    • 著者名/発表者名
      Sato Naoki
    • 雑誌名

      Plasma Physics and Controlled Fusion

      巻: 61 ページ: 125013~125013

    • DOI

      doi.org/10.1088/1361-6587/ab5001

    • 査読あり
  • [雑誌論文] Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems2019

    • 著者名/発表者名
      Sato Naoki、Yamada Michio
    • 雑誌名

      Journal of Fluid Mechanics

      巻: 876 ページ: 896~911

    • DOI

      doi.org/10.1017/jfm.2019.565

    • 査読あり
  • [雑誌論文] Degenerate Laplacian describing topologically constrained diffusion: helicity constraint as an alternative to ellipticity2019

    • 著者名/発表者名
      Sato N、Yoshida Z
    • 雑誌名

      Journal of Physics A: Mathematical and Theoretical

      巻: 52 ページ: 355202~355202

    • DOI

      doi.org/10.1088/1751-8121/ab306a

    • 査読あり
  • [雑誌論文] Local representation and construction of Beltrami fields II.solenoidal Beltrami fields and ideal MHD equilibria2019

    • 著者名/発表者名
      Sato N.、Yamada M.
    • 雑誌名

      Physica D: Nonlinear Phenomena

      巻: 400 ページ: 132142~132142

    • DOI

      doi.org/10.1016/j.physd.2019.06.008

    • 査読あり
  • [雑誌論文] Beltrami operators and their application to constrained diffusion in Beltrami fields2019

    • 著者名/発表者名
      Sato N
    • 雑誌名

      Journal of Physics A: Mathematical and Theoretical

      巻: 52 ページ: 235203~235203

    • DOI

      doi.org/10.1088/1751-8121/ab1cdc

    • 査読あり
  • [学会発表] Existence of ideal magnetofluid equilibria without continuous Euclidean symmetries2019

    • 著者名/発表者名
      Naoki Sato
    • 学会等名
      Phenomena and Mathematical Models 2019, Hotel Plaza Annex Yokote, Yokote, Japan
  • [学会発表] Local Clebsch parametrization of Beltrami equilibria2019

    • 著者名/発表者名
      Naoki Sato, Michio Yamada, and Zensho Yoshida
    • 学会等名
      3rd Asia Pacific Conference on Plasma Physics, Hefei, China
    • 国際学会 / 招待講演
  • [学会発表] Superharmonic instability of nonlinear traveling wave solutions in Hamiltonian systems2019

    • 著者名/発表者名
      Naoki Sato and Michio Yamada
    • 学会等名
      Mathematical Aspects of Nonlinear Waves and their Applications, Kyoto University, Kyoto, Japan
  • [学会発表] Singular solutions of the Euler equations by Clebsch parametrization of Beltrami fields2019

    • 著者名/発表者名
      Naoki Sato and Michio Yamada
    • 学会等名
      C000063, The 65th National Congress of Theoretical and Applied Mechanics, Hokkaido University, Sapporo, Japan
  • [学会発表] Singular and asymmetric Beltrami field solutions of the ideal MHD equations2019

    • 著者名/発表者名
      Naoki Sato, Michio Yamada, and Zensho Yoshida
    • 学会等名
      mini-course/workshop on the application of computational mathematics to plasma physics, Mathematical Sciences Institute, Australian National University, Canberra, Australia
    • 国際学会 / 招待講演
  • [学会発表] An introduction to Beltrami fields in plasma and fluid dynamics2019

    • 著者名/発表者名
      Naoki Sato
    • 学会等名
      mini-course/workshop on the application of computational mathematics to plasma physics, Mathematical Sciences Institute, Australian National University, Canberra, Australia
    • 国際学会 / 招待講演
  • [備考] Google Scholar

    • URL

      https://scholar.google.com/citations?user=VOI6Ej0AAAAJ&hl=en

  • [備考] Homepage

    • URL

      https://naokisato.work

  • [備考] Researchmap

    • URL

      https://researchmap.jp/naoki_sato

URL: 

公開日: 2021-01-27  

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