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Asymptotic reflection of a self-propelled particle from a boundary wall

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Abstract

Exact expressions for the relationship between angles of incidence and reflection from a boundary wall in nonlinear dissipative particle models are extremely rare. Here, we study a particle model of a camphor disk floating on water in a low speed limit, for which the model was derived. We begin with a very rough and inaccurate approximation, based in part on a Hamiltonian limit of the model, and then, using symmetry arguments supported by a series of experiments, show that this rough approximation can likely be repaired by introducing a factor dependent upon model parameters, then extract the full parameter dependence of this factor, and finally conjecture a simple and exact asymptotic relationship between the angles of incidence and reflection. There are reasons to believe that this asymptotic relationship may be universal for such models in their corresponding limits.

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Number 19K03626 and MEXT Promotion of Distinctive Joint Research Center Program Grant Number JPMXP0620335886.

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Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Tomoyuki Miyaji

  2. Hosei University, Tokyo, Japan

    Robert Sinclair

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  1. Tomoyuki Miyaji
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  2. Robert Sinclair
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Correspondence to Tomoyuki Miyaji.

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Dedicated to the memory of Professor Masayasu Mimura.

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Miyaji, T., Sinclair, R. Asymptotic reflection of a self-propelled particle from a boundary wall. Japan J. Indust. Appl. Math. 41, 269–295 (2024). https://doi.org/10.1007/s13160-023-00602-w

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  • DOI: https://doi.org/10.1007/s13160-023-00602-w

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