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Statistical mechanics of generalized conservative systems: self-organization by non-integrable topological constraints and non-elliptic diffusion processes

Research Project

Project/Area Number 18J01729
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section国内
Research Field Mathematical physics/Fundamental condensed matter physics
Research InstitutionKyoto University

Principal Investigator

佐藤 直木  京都大学, 数理解析研究所, 特別研究員(PD)

Project Period (FY) 2018-04-25 – 2021-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
KeywordsMetriplectic brackets / Euler Equations / Stellarator / Degenerate-Elliptic PDE / Elliptic-Hyperbolic PDE / Beltrami fields / Up-Hill Diffusion / Topological Constraints / Beltrami Fields / Darboux Theorem / Almost Poisson Operators
Outline of Annual Research Achievements

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.

Research Progress Status

翌年度、交付申請を辞退するため、記入しない。

Strategy for Future Research Activity

翌年度、交付申請を辞退するため、記入しない。

Report

(2 results)
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • Research Products

    (24 results)

All 2019 2018 Other

All Journal Article (6 results) (of which Peer Reviewed: 6 results) Presentation (14 results) (of which Int'l Joint Research: 8 results,  Invited: 4 results) Remarks (4 results)

  • [Journal Article] Existence of ideal magnetofluid equilibria without continuous Euclidean symmetries2019

    • Author(s)
      Sato Naoki
    • Journal Title

      Plasma Physics and Controlled Fusion

      Volume: 61 Issue: 12 Pages: 125013-125013

    • DOI

      10.1088/1361-6587/ab5001

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems2019

    • Author(s)
      Sato Naoki、Yamada Michio
    • Journal Title

      Journal of Fluid Mechanics

      Volume: 876 Pages: 896-911

    • DOI

      10.1017/jfm.2019.565

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Degenerate Laplacian describing topologically constrained diffusion: helicity constraint as an alternative to ellipticity2019

    • Author(s)
      Sato N、Yoshida Z
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 52 Issue: 35 Pages: 355202-355202

    • DOI

      10.1088/1751-8121/ab306a

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Local representation and construction of Beltrami fields II.solenoidal Beltrami fields and ideal MHD equilibria2019

    • Author(s)
      Sato N.、Yamada M.
    • Journal Title

      Physica D: Nonlinear Phenomena

      Volume: 400 Pages: 132142-132142

    • DOI

      10.1016/j.physd.2019.06.008

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Beltrami operators and their application to constrained diffusion in Beltrami fields2019

    • Author(s)
      Sato N
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 52 Issue: 23 Pages: 235203-235203

    • DOI

      10.1088/1751-8121/ab1cdc

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Local representation and construction of Beltrami fields2019

    • Author(s)
      N. Sato and M. Yamada
    • Journal Title

      Physica D: Nonlinear Phenomena

      Volume: 391 Pages: 8-16

    • DOI

      10.1016/j.physd.2019.02.003

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Presentation] Existence of ideal magnetofluid equilibria without continuous Euclidean symmetries2019

    • Author(s)
      Naoki Sato
    • Organizer
      Phenomena and Mathematical Models 2019, Hotel Plaza Annex Yokote, Yokote, Japan
    • Related Report
      2019 Annual Research Report
  • [Presentation] Local Clebsch parametrization of Beltrami equilibria2019

    • Author(s)
      Naoki Sato, Michio Yamada, and Zensho Yoshida
    • Organizer
      3rd Asia Pacific Conference on Plasma Physics, Hefei, China
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Superharmonic instability of nonlinear traveling wave solutions in Hamiltonian systems2019

    • Author(s)
      Naoki Sato and Michio Yamada
    • Organizer
      Mathematical Aspects of Nonlinear Waves and their Applications, Kyoto University, Kyoto, Japan
    • Related Report
      2019 Annual Research Report
  • [Presentation] Singular solutions of the Euler equations by Clebsch parametrization of Beltrami fields2019

    • Author(s)
      Naoki Sato and Michio Yamada
    • Organizer
      C000063, The 65th National Congress of Theoretical and Applied Mechanics, Hokkaido University, Sapporo, Japan
    • Related Report
      2019 Annual Research Report
  • [Presentation] Singular and asymmetric Beltrami field solutions of the ideal MHD equations2019

    • Author(s)
      Naoki Sato, Michio Yamada, and Zensho Yoshida
    • Organizer
      mini-course/workshop on the application of computational mathematics to plasma physics, Mathematical Sciences Institute, Australian National University, Canberra, Australia
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] An introduction to Beltrami fields in plasma and fluid dynamics2019

    • Author(s)
      Naoki Sato
    • Organizer
      mini-course/workshop on the application of computational mathematics to plasma physics, Mathematical Sciences Institute, Australian National University, Canberra, Australia
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Local representation and construction of Beltrami field solutions of the Euler equations2019

    • Author(s)
      N. Sato and M. Yamada
    • Organizer
      The Physical Society of Japan 74th annual meeting
    • Related Report
      2018 Annual Research Report
  • [Presentation] Beltrami fields: representation, construction, and asymmetric equilibria2018

    • Author(s)
      N. Sato and M. Yamada
    • Organizer
      現象と数理モデル2018
    • Related Report
      2018 Annual Research Report
  • [Presentation] Statistical mechanics of topologically constrained systems: application to self-organizing diffusion in plasmas2018

    • Author(s)
      N. Sato and Z. Yoshida
    • Organizer
      2nd Asia Pacific Conference on Plasma Physics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Statistical mechanics of Kida vortex: an effect of singularity in noncanonical Hamiltonian system2018

    • Author(s)
      N. Sato, Z. Yoshida, and P. J. Morrison
    • Organizer
      15th International Conference on Flow Dynamics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Invariant measure and statistical mechanics of 3D Lie-Poisson algebras: structures created by the singular point of Bianchi class B2018

    • Author(s)
      N. Sato, Z. Yoshida, and P. J. Morrison
    • Organizer
      The Physical Society of Japan 73rd annual meeting
    • Related Report
      2018 Annual Research Report
  • [Presentation] Local representation and construction of Beltrami fields2018

    • Author(s)
      N. Sato and M. Yamada
    • Organizer
      Hamiltonian systems, from topology to applications through analysis I
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Hamiltonian and non-Hamiltonian reductions of conservative dynamics: structures created by degenerate, singular, and finite helicity field-tensors2018

    • Author(s)
      N. Sato and Z. Yoshida
    • Organizer
      The 32nd Colloquium on Group Theoretical Methods in Physics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Multiregion relaxed MHD toroidal states with flow2018

    • Author(s)
      R. L. Dewar, Z. S. Qu, N. Sato, S. R. Hudson, and M. J. Hole
    • Organizer
      19th International Congress on Plasma Physics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Remarks] Google Scholar

    • URL

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

    • Related Report
      2019 Annual Research Report
  • [Remarks] Homepage

    • URL

      https://naokisato.work

    • Related Report
      2019 Annual Research Report
  • [Remarks] Researchmap

    • URL

      https://researchmap.jp/naoki_sato

    • Related Report
      2019 Annual Research Report
  • [Remarks] Google scholar profile

    • URL

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

    • Related Report
      2018 Annual Research Report

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Published: 2018-05-01   Modified: 2024-03-26  

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