2023 Fiscal Year Final Research Report
A billiard problem arising from self-propelling particles
| Project/Area Number |
19K03626
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
SINCLAIR Robert 法政大学, 経済学部, 客員教授 (50423744)
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | 自己駆動粒子 / 数理モデリング / ビリヤード問題 / 力学系 / 微分方程式 / 数値シミュレーション |
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| Outline of Final Research Achievements |
We studied the billiard-like motion of a single self-propelled particle that is asymptotically in constant velocity linear motion inside a domain and interacts repulsively with the boundary. We have conducted mathematical and numerical analyses on three different mathematical models: partial differential equation models, ordinary differential equation models, and discrete-time dynamical system models. The published results concern the reflection of particles in the ordinary differential equation model. We have proved under certain assumptions that the angle of reflection is greater than that of incidence. Numerical results suggest that this assumption can be relaxed, but the proof remains difficult. In addition, we have carried out detailed numerical experiments and have obtained a conjecture about the specific form of the functional relationship between the angle of incidence and that of reflection in the limit of very slow particle motion.
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| Free Research Field |
非線型解析
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| Academic Significance and Societal Importance of the Research Achievements |
本研究は水面に浮かぶ樟脳円板の運動を主な動機付けとしてきたが，入射角と反射角の関係に関して得られた成果は，偏微分方程式モデルから分岐理論による縮約方程式として導かれたモデルの研究を通して得たものである．そのため，特定の方程式や系にとどまらず，同様の振る舞いを示す別の系においても同様のことが成り立つ普遍性があると予想する．我々の提出した予想を数学的に証明する試みが，自己駆動粒子の数理モデルに対する数学解析をさらに促進することを期待する．
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