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

Durey, M. , Milewski, P. A. and Bush, J. W. M. (2018) Dynamics, emergent statistics, and the mean-pilot-wave potential of walking droplets. Chaos: An Interdisciplinary Journal of Nonlinear Science, 28(9), 096108. (doi: 10.1063/1.5030639) (PMID:30278646)

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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Durey, Dr Matthew
• Authors: Durey, M., Milewski, P. A., and Bush, J. W. M.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Chaos: An Interdisciplinary Journal of Nonlinear Science
• Publisher: American Institute of Physics
• ISSN: 1054-1500
• ISSN (Online): 1089-7682
• Published Online: 18 September 2018
• Copyright Holders: Copyright © 2018 The Authors
• First Published: First published in Chaos: An Interdisciplinary Journal of Nonlinear Science 28(9): 096108
• Publisher Policy: Reproduced under a Creative Commons License