2016 Fiscal Year Annual Research Report
Non-Equilibrium Statistical Mechanics with Topological Constraints: Thermodynamics and Entropy Production of Self-Organized Turbulence
| Project/Area Number |
16J01486
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| Research Institution | The University of Tokyo |
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Principal Investigator |
佐藤 直木 東京大学, 大学院新領域創成科学研究科, 特別研究員(DC2)
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| Project Period (FY) |
2016-04-22 – 2018-03-31
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| Keywords | Conservative Dynamics / Topological Constraints / Entropy Measure / Self-Organization / Hamiltonian Mechanics / Almost Poisson Algebra / Invariant Measure / Jacobi Identity |
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| Outline of Annual Research Achievements |
Aim of the present study is to investigate self-organization phenomena caused by topological constraints (geometrical constraints imposed on the phase space). We achieved the following results:
１、As an example of self-organization caused by integrable constraints, we studied the creation of a radiation belt. Such process can be understood by formulating a theory of diffusion on a curved metric induced by the so called adiabatic invariants of magnetized particles (which act as constraints). We showed that, on the curved metric, the creation of the radiation belt is consistent with the second law of thermodynamics.
２、We generalized the theory to the whole class of integrable constraints, i.e. to non-canonical Hamiltonian systems. We formulated their entropy measure, and showed the consistency with the second law of thermodynamics. This result demonstrates that self-organization driven by topological constraints is consistent with the maximum entropy principle.
３、We investigated self-organization driven by non-integrable constraints. Systems with such constraints are described by almost Poisson operators. We categorized almost Poisson operators according to their geometrical properties, and identified three new classes: conformal, measure preserving, and Beltrami. We then proved a series of theorems concerning the equilibrium probability distribution of systems endowed with such operators, and showed that non-integrability introduces a new type of self-organization caused by the vorticity of the operators.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
We have achieved the main objectives of research for the first year of investigation. Indeed, we clarified the relationship between topological constraints, entropy production, and thermodynamics for the specific cases discussed in the original plan. In addition, we were able to construct a comprehensive and general formulation of the statistical mechanics of conservative systems affected by both integrable and non-integrable topological constraints, and thus advance further the present study. This novel formulation is expected to have applications in plasma physics, analytical mechanics, statistical mechanics, pure and applied mathematics.
At the same time, we have started an international collaboration that is focused on the effects of topological constraints on plasma and fluid relaxation. By introducing a novel potential representation of the ideal Euler flow, we derived a new set of equations describing a strongly relaxed plasma dynamics. This parallel investigation should have practical applications to the numerical and experimental modelling of fusion plasmas in tokamaks and other magnetic confinement devices.
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| Strategy for Future Research Activity |
The current research plan for the second year of investigation is organized as follows:
１、Introduce a rigorous measure for the vorticity of almost Poisson operators and show how such measure determines the self-organized structures and the associated entropy measure. This step should complete the objective of the original research project.
２、Study existence, uniqueness, and convergence properties of the solution to the non-elliptic partial differential equation arising in the case of a purely diffusive system with non-integrable topological constraints.
３、Enhance and advance the current international collaboration and apply the developed relaxation model to the analysis of stationary plasma flow in magnetic confinement devices.
４、Put additional effort to spread the results obtained within the present investigation by participating to international meetings and contributing to the scientific literature.
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