The invariant measure of a walking droplet in hydrodynamic pilot-wave theory
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866916491223367680 |
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| author | Nguyen, Hung D. Oza, Anand U. |
| author_facet | Nguyen, Hung D. Oza, Anand U. |
| contents | We study the long time statistics of a walker in a hydrodynamic pilot-wave system, which is a stochastic Langevin dynamics with an external potential and memory kernel. While prior experiments and numerical simulations have indicated that the system may reach a statistically steady state, its long-time behavior has not been studied rigorously. For a broad class of external potentials and pilot-wave forces, we construct the solutions as a dynamics evolving on suitable path spaces. Then, under the assumption that the pilot-wave force is dominated by the potential, we demonstrate that the walker possesses a unique statistical steady state. We conclude by presenting an example of such an invariant measure, as obtained from a numerical simulation of a walker in a harmonic potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_11767 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The invariant measure of a walking droplet in hydrodynamic pilot-wave theory Nguyen, Hung D. Oza, Anand U. Probability Mathematical Physics We study the long time statistics of a walker in a hydrodynamic pilot-wave system, which is a stochastic Langevin dynamics with an external potential and memory kernel. While prior experiments and numerical simulations have indicated that the system may reach a statistically steady state, its long-time behavior has not been studied rigorously. For a broad class of external potentials and pilot-wave forces, we construct the solutions as a dynamics evolving on suitable path spaces. Then, under the assumption that the pilot-wave force is dominated by the potential, we demonstrate that the walker possesses a unique statistical steady state. We conclude by presenting an example of such an invariant measure, as obtained from a numerical simulation of a walker in a harmonic potential. |
| title | The invariant measure of a walking droplet in hydrodynamic pilot-wave theory |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2210.11767 |