| Project/Area Number |
16K13764
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Basic analysis
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| Research Institution | Kyushu University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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| Keywords | 無限粒子系 / クーロンポテンシャル / 相転移現象 / 均質化 / 行列式測度 / 遠距離強相互作用 / ケプラー問題 |
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| Outline of Final Research Achievements |
We replace the particles periodically placed on the two-dimensional lattice points except that only one particle removed from the origin (these are called environment particles), and consider that another Brownian particle is moving in the plane effected from the environment particle system by the Coulomb interaction potential with inverse temperature β=2. We investigated the diffusive limit when the particle was moving while receiving repulsive force at the Coulomb potential. And it was proved that phase transition occurred about β in this situation.
In the rigid Coulomb's equilibrium distribution, the only one that is now composed is the Ginibre point process. The Ginibre point process is a typical determinant measure. In the present study, we proved that in continuous space, the tail events of all determinant measures are trivial. Our result is an extension to the continuous space which was conventionally known only in the discrete space.
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| Academic Significance and Societal Importance of the Research Achievements |
d次元空間でd次元クーロンポテンシャルによって相互作用する無限個の粒子を厳密クーロン無限粒子系とよぶ。その平衡分布は、次元 d = 2 かつ逆温度β=2の場合が、現在知られている唯一の例であり、Ginibre 点過程と呼ばれる。クーロンポテンシャルの遠方での相互作用の強烈さのために、厳密クーロン無限粒子系は、通常のギッブス点過程やポアソン点過程とは、非常に異なる様相を持つと予想される。この研究では、その例としてクーロン力で影響されるランダム環境中の粒子の均質化問題に関する有効係数についての相転移現象を証明した。
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