2018 Fiscal Year Annual Research Report
Statistical mechanics of generalized conservative systems: self-organization by non-integrable topological constraints and non-elliptic diffusion processes
| Project/Area Number |
18J01729
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| Research Institution | Kyoto University |
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Principal Investigator |
佐藤 直木 京都大学, 数理解析研究所, 特別研究員(PD)
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| Project Period (FY) |
2018-04-25 – 2021-03-31
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| Keywords | Degenerate-Elliptic PDE / Up-Hill Diffusion / Topological Constraints / Beltrami Fields / Darboux Theorem / Almost Poisson Operators / Euler Equations |
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| 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、We proved the existence of weak solutions to the boundary value problem for the orthogonal Laplacian, a second order degenerate elliptic partial differential operator representing the diffusion operator of mechanical systems subject to topological constraints. The solution is unique if the constraints are non-integrable.
2、We formulated the statistical theory of Beltrami operators, a class of antisymmetric operators that generalize the Poisson operator of noncanonical Hamiltonian mechanics to certain systems subject to non-integrable constraints.
3、We studied the existence of Beltrami operators in 3 dimensions, and obtained a local representation theorem and a construction method for Beltrami fields. This result provides an effective method to construct analytic solutions of the Euler and Navier-Stokes equations.
4、We investigated self-organization caused by singularities (rank drops) in the Poisson operator of 3-dimensional Lie-Poisson algebras, and calculated the resulting distortion of the equilibrium distribution function.
5、We proved that translationally symmetric traveling wave solutions to canonical Hamiltonian systems subject to superharmonic perturbations exchange their stability at the extrema of energy with respect to wave speed.
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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 achieved the main objectives of the first year of investigation. First, we proved the existence of weak solutions to the boundary value problem for the second order degenerate-elliptic partial differential operator (orthogonal Laplacian) describing diffusion processes in systems with topological constraints. This result clarifies the relationship between the integrability of a topological constraint and the nature of the solution to the associated diffusion equation, and thus fulfills the main purpose of the original plan. Furthermore, we studied statistical properties of Beltrami operators, which generalize the Poisson operator of Hamiltonian mechanics to a class of systems with non-integrable constraints and mathematically determine the form of the diffusion operator above. In particular, we derived the entropy measure associated to Beltrami operators, and showed that the standard results of statistical mechanics can be extended to this class of dynamics. In addition, we examined the existence of Beltrami fields, i.e. Beltrami operators in 3 real dimensions, and proved a local representation theorem that enables the construction of analytical solutions to the Euler and Navier-Stokes equations. These results, which were corroborated through numerical simulations, fulfill the second objective of the original plan. Finally, the study on the effect of singularities on the equilibrium distribution in 3-dimensional Hamiltonian systems, and the stability analysis of traveling wave solutions represent additional achievements that were not expected in the original plan.
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| Strategy for Future Research Activity |
The research plan for the second year of investigation is organized as follows:
1、It has emerged that Beltrami operators represent the key class of dynamics for the understanding of the statistical properties of systems endowed with non-integrable topological constraints. We plan to conduct additional research on the existence of such operators in 3 real dimensions, and to extend the local theory developed so far to a global one in order to enforce boundary conditions.
2、Due to the occurrence of Beltrami fields in fluid systems, we plan to apply the present theory to the study of steady equilibria of the Euler equations, and to decaying solutions of the Navier-Stokes equations. Furthermore, we plan to conduct research related to the Euler equation in collaboration with overseas institutions.
3、The results regarding the existence and uniqueness of solution to the boundary value problem associated to the orthogonal Laplacian require additional investigation in the following directions. First, we plan to study the function space of the solutions and the associated norm. Secondly, we plan to generalize the theory to a wider class of second order degenerate elliptic partial differential equations.
4、As an application to fluid dynamics of the theory of diffusion in topologically constrained systems developed so far, we plan to investigate the stability of steady solutions of the Euler equation by considering the evolution of perturbations as a non-local diffusion problem in Fourier space.
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