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

Matthew Durey,John W M Bush,Paul A Milewski

doi:10.1063/1.5030639

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.

Highlights

A millimetric droplet may bounce on the surface of a vertically vibrating bath of the same fluid; the thin air layer separating the droplet from the bath during impact prevents coalescence.[1,2] Each impact excites a field of temporally decaying Faraday waves, whose longevity depends on the reduced acceleration = Aω02/g, where A is the shaking amplitude, ω0/(2π ) is the frequency, and g is the gravitational acceleration
When the droplet is subject to a central force with its motion confined to a line, we rationalize a number of regimes, including periodic quantized oscillations, chaotic motion, and the emergence of wavelike statistics
We demonstrate that the mean-pilot-wave potential has a controlling influence on the droplet’s dynamics at high vibrational forcing, where the resultant droplet motion is similar to a random walk

Summary

Introduction

The bouncing may destabilize to horizontal “walking” across the bath, whereby the droplet is propelled at each impact by the slope of its associated Faraday wave field3 [see Fig. 1(a)]. The decay time of the Faraday waves increases with for < F, where the Faraday threshold F is the critical vibrational acceleration at which Faraday waves arise in the absence of a droplet. This decay time results in a “path-memory” of previous impacts, where the memory timescale is inversely proportional to the proximity of the Faraday threshold F.4. This decay time results in a “path-memory” of previous impacts, where the memory timescale is inversely proportional to the proximity of the Faraday threshold F.4 The resulting dynamics are similar in many respects to the pilot-wave dynamics envisaged by de Broglie as a physical framework for understanding quantum mechanics.[5]

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Conclusion