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Dynamics, emergent statistics, and the mean-pilot-wave potential of walking droplets

Author(s)
Durey, Matthew; Milewski, Paul A.; Bush, John W. M.
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Creative Commons Attribution 4.0 International license https://creativecommons.org/licenses/by/4.0/
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Abstract
A millimetric droplet may bounce and self-propel on the surface of a vertically vibrating bath, where its horizontal "walking" motion is induced by repeated impacts with its accompanying Faraday wave field. For ergodic long-time dynamics, we derive the relationship between the droplet's stationary statistical distribution and its mean wave field in a very general setting. We then focus on the case of a droplet subjected to a harmonic potential with its motion confined to a line. By analyzing the system's periodic states, we reveal a number of dynamical regimes, including those characterized by stationary bouncing droplets trapped by the harmonic potential, periodic quantized oscillations, chaotic motion and wavelike statistics, and periodic wave-trapped droplet motion that may persist even in the absence of a central force. We demonstrate that as the vibrational forcing is increased progressively, the periodic oscillations become chaotic via the Ruelle-Takens-Newhouse route. We rationalize the role of the local pilot-wave structure on the resulting droplet motion, which is akin to a random walk. We characterize the emergence of wavelike statistics influenced by the effective potential that is induced by the mean Faraday wave field.
Date issued
2018-09
URI
https://hdl.handle.net/1721.1/123507
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Chaos: An Interdisciplinary Journal of Nonlinear Science
Publisher
American Institute of Physics
Citation
Durey, Matthew et al. "Dynamics, emergent statistics, and the mean-pilot-wave potential of walking droplets." Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 9 (September 2018): 096108 © 2018 Author(s).
Version: Final published version
ISSN
1054-1500
1089-7682

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