Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1504.00484 · Physical Review E

Strings of droplets propelled by coherent waves

Boris Filoux, Maxime Hubert, Nicolas Vandewalle

Published Oct 29, 2015• 32 citations
arXivHTMLPDF

Abstract

Bouncing walking droplets possess fascinating properties due to their peculiar wave-particle interaction leading to unexpected quantumlike behaviors. We propose a study consisting in droplets walking along annular cavities. We show that, in this geometry, they spontaneously form a string of synchronized bouncing droplets that share a common coherent wave propelling the group at a speed faster than single walkers. The formation of this coherent wave and the collective droplet behaviors are captured by a model. Those are at the opposite of the ones found in two-dimensional geometries. Our results shed light on walking dynamics.

DOI
10.1103/physreve.92.041004

Related papers

The pilot-wave dynamics of walking droplets

Daniel M. Harris, John W. M. Bush

A millimetric droplet can be induced to bounce on the surface of a fluid bath by vibrating the bath near the droplet's resonant frequency (Figure 1(a) ). [superscript 1–3] The localized field of Faraday waves excited by the bouncing droplet can cause it to propel itself laterally across the surface, moving in resonance with its guiding wave field (Figure 1(b) ). [superscript 4,5] These walking droplets, or “walkers,” generally move in a straight line at constant speed; however, they can be diverted through interaction with boundaries or external forces. This hydrodynamic system represents a macroscopic realization of the pilot-wave theory of quantum dynamics proposed by Louis de Broglie, according to which microscopic particles are propelled through a resonant interaction with a wave field generated by the particle's internal vibration. [superscript 6] Coincidentally, it exhibits many behaviors once thought to be exclusive to the microscopic quantum realm, including single-particle diffraction, [superscript 7] tunneling, [superscript 8] quantized orbits, [superscript 9] and orbital-level splitting. [superscript 10]

Physics of Fluids•Sep 1, 2013•22 citations

One-dimensional pilot-wave dynamics of walking droplets

S.J.H. Koerhuis

Pilot-wave dynamics of walking droplets is a field in hydrodynamics that has been studied extensively for its interesting features previously thought to be found only in quantum mechanics. The walking droplets in combination with their pilot-wave form a spatially delocalized entity called a walker, wherefrom quantum-like features arise. \n\nThe behavior and dynamics of the droplet have been described theoretically, where the walker moves in a two-dimensional (plane) bath. We start the theoretical development of an analogous one-dimensional version of this theory that describes the walker's behavior in an one-dimensional (long and narrow) bath. We set up the model basis for the equation of motion of the vertically bouncing droplet by looking at a quasi-static droplet in our one-dimensional system. \n\nOur goal is to develop a theory that eventually describes and rationalizes the dynamics and statistical behavior of a walker in our one-dimensional setting. We predict that the simplification to an one-dimensional setting will make the theory behind the behavior of the walker system more understandable. Hence, the theory, which is the basis for explaining the emerging quantum-like features, will eventually shed some light on what is and what is not fundamental to quantum mechanics.\n\nSpecifically, the surface- and potential energy of a quasi-static droplet are calculated and expressed in an harmonic decomposition of the droplet's shape. The expansion coefficients that arise in the harmonic decomposition are calculated by performing an averaging method on the potential energy and minimizing the total energy of a droplet resting on a rigid surface. The expansion coefficients are shown to accurately describe the droplet's shape for a range of Bond numbers (a dimensionless number representing the relative magnitude of gravity to surface tension per unit area), where the range depends on the contact area that the droplet makes with the underlying surface. These results can be used to further develop the one-dimensional walker theory.

•Jan 1, 2020•0 citations

Walkers in a wave field with memory

Olivier Devauchelle, É. Lajeunesse, François James +2 more

Couder et al. [1] discovered that the behavior of droplets bouncing on a vibrated bath mimics a variety of quantum phenomena, through their coupling with a Faraday wave. Here, we introduce a continuous model inspired from this experiment, which encapsulates its essential features into a system of three simple differential equations. We show, numerically and analytically, the existence of self-propelled walkers in this framework. Finally, making use of the model’s familiar formulation, we suggest that a particle coupled with a propagating wave, instead of a standing one, behaves much like a bouncing droplet.

Comptes Rendus Mécanique•Nov 16, 2020•4 citations

Dynamics and statistics of wave-particle interactions in a confined geometry

Tristan Gilet

A walker is a droplet bouncing on a liquid surface and propelled by the waves that it generates. This macroscopic wave-particle association exhibits behaviors reminiscent of quantum particles. This article presents a toy model of the coupling between a particle and a confined standing wave. The resulting two-dimensional iterated map captures many features of the walker dynamics observed in different configurations of confinement. These features include the time decomposition of the chaotic trajectory in quantized eigenstates and the particle statistics being shaped by the wave. It shows that deterministic wave-particle coupling expressed in its simplest form can account for some quantumlike behaviors.

Physical Review E•Nov 21, 2014•56 citations

Wavelike statistics from pilot-wave dynamics in a circular corral

Daniel M. Harris, Julien Moukhtar, Emmanuel Fort +2 more

Bouncing droplets can self-propel laterally along the surface of a vibrated fluid bath by virtue of a resonant interaction with their own wave field. The resulting walking droplets exhibit features reminiscent of microscopic quantum particles. Here we present the results of an experimental investigation of droplets walking in a circular corral. We demonstrate that a coherent wavelike statistical behavior emerges from the complex underlying dynamics and that the probability distribution is prescribed by the Faraday wave mode of the corral. The statistical behavior of the walking droplets is demonstrated to be analogous to that of electrons in quantum corrals.

Physical Review E•Jul 10, 2013•173 citations

Walking droplets correlated at a distance

André Nachbin

Bouncing fluid droplets can walk on the surface of a vibrating bath forming a wave-particle association. Walking droplets have many quantum-like features. Research efforts are continuously exploring quantum analogues and respective limitations. Here, we demonstrate that two oscillating particles (millimetric droplets) confined to separate potential wells exhibit correlated dynamical features, even when separated by a large distance. A key feature is the underlying wave mediated dynamics. The particles' phase space dynamics is given by the system as a whole and cannot be described independently. Numerical phase space histograms display statistical coherence; the particles' intricate distributions in phase space are statistically indistinguishable. However, removing one particle changes the phase space picture completely, which is reminiscent of entanglement. The model here presented also relates to nonlinearly coupled oscillators where synchronization can break out spontaneously. The present oscillator-coupling is dynamic and can change intensity through the underlying wave field as opposed to, for example, the Kuramoto model where the coupling is pre-defined. There are some regimes where we observe phase-locking or, more generally, regimes where the oscillators are statistically indistinguishable in phase-space, where numerical histograms display their (mutual) most likely amplitude and phase.

Chaos An Interdisciplinary Journal of Nonlinear Science•Sep 1, 2018•28 citations
Bouncing walking droplets possess fascinating properties due to their peculiar wave-particle interaction leading to unexpected quantumlike behaviors. We propose a study consisting in droplets walking along annular cavities. We show that, in this geometry, they spontaneously form a string of synchronized bouncing droplets that share a common coherent wave propelling the group at a speed faster than single walkers. The formation of this coherent wave and the collective droplet behaviors are captured by a model. Those are at the opposite of the ones found in two-dimensional geometries. Our results shed light on walking dynamics.
A millimetric droplet can be induced to bounce on the surface of a fluid bath by vibrating the bath near the droplet's resonant frequency (Figure 1(a) ). [superscript 1–3] The localized field of Faraday waves excited by the bouncing droplet can cause it to propel itself laterally across the surface, moving in resonance with its guiding wave field (Figure 1(b) ). [superscript 4,5] These walking droplets, or “walkers,” generally move in a straight line at constant speed; however, they can be diverted through interaction with boundaries or external forces. This hydrodynamic system represents a macroscopic realization of the pilot-wave theory of quantum dynamics proposed by Louis de Broglie, according to which microscopic particles are propelled through a resonant interaction with a wave field generated by the particle's internal vibration. [superscript 6] Coincidentally, it exhibits many behaviors once thought to be exclusive to the microscopic quantum realm, including single-particle diffraction, [superscript 7] tunneling, [superscript 8] quantized orbits, [superscript 9] and orbital-level splitting. [superscript 10]
Pilot-wave dynamics of walking droplets is a field in hydrodynamics that has been studied extensively for its interesting features previously thought to be found only in quantum mechanics. The walking droplets in combination with their pilot-wave form a spatially delocalized entity called a walker, wherefrom quantum-like features arise. \n\nThe behavior and dynamics of the droplet have been described theoretically, where the walker moves in a two-dimensional (plane) bath. We start the theoretical development of an analogous one-dimensional version of this theory that describes the walker's behavior in an one-dimensional (long and narrow) bath. We set up the model basis for the equation of motion of the vertically bouncing droplet by looking at a quasi-static droplet in our one-dimensional system. \n\nOur goal is to develop a theory that eventually describes and rationalizes the dynamics and statistical behavior of a walker in our one-dimensional setting. We predict that the simplification to an one-dimensional setting will make the theory behind the behavior of the walker system more understandable. Hence, the theory, which is the basis for explaining the emerging quantum-like features, will eventually shed some light on what is and what is not fundamental to quantum mechanics.\n\nSpecifically, the surface- and potential energy of a quasi-static droplet are calculated and expressed in an harmonic decomposition of the droplet's shape. The expansion coefficients that arise in the harmonic decomposition are calculated by performing an averaging method on the potential energy and minimizing the total energy of a droplet resting on a rigid surface. The expansion coefficients are shown to accurately describe the droplet's shape for a range of Bond numbers (a dimensionless number representing the relative magnitude of gravity to surface tension per unit area), where the range depends on the contact area that the droplet makes with the underlying surface. These results can be used to further develop the one-dimensional walker theory.
Couder et al. [1] discovered that the behavior of droplets bouncing on a vibrated bath mimics a variety of quantum phenomena, through their coupling with a Faraday wave. Here, we introduce a continuous model inspired from this experiment, which encapsulates its essential features into a system of three simple differential equations. We show, numerically and analytically, the existence of self-propelled walkers in this framework. Finally, making use of the model’s familiar formulation, we suggest that a particle coupled with a propagating wave, instead of a standing one, behaves much like a bouncing droplet.
A walker is a droplet bouncing on a liquid surface and propelled by the waves that it generates. This macroscopic wave-particle association exhibits behaviors reminiscent of quantum particles. This article presents a toy model of the coupling between a particle and a confined standing wave. The resulting two-dimensional iterated map captures many features of the walker dynamics observed in different configurations of confinement. These features include the time decomposition of the chaotic trajectory in quantized eigenstates and the particle statistics being shaped by the wave. It shows that deterministic wave-particle coupling expressed in its simplest form can account for some quantumlike behaviors.
Bouncing droplets can self-propel laterally along the surface of a vibrated fluid bath by virtue of a resonant interaction with their own wave field. The resulting walking droplets exhibit features reminiscent of microscopic quantum particles. Here we present the results of an experimental investigation of droplets walking in a circular corral. We demonstrate that a coherent wavelike statistical behavior emerges from the complex underlying dynamics and that the probability distribution is prescribed by the Faraday wave mode of the corral. The statistical behavior of the walking droplets is demonstrated to be analogous to that of electrons in quantum corrals.
Bouncing fluid droplets can walk on the surface of a vibrating bath forming a wave-particle association. Walking droplets have many quantum-like features. Research efforts are continuously exploring quantum analogues and respective limitations. Here, we demonstrate that two oscillating particles (millimetric droplets) confined to separate potential wells exhibit correlated dynamical features, even when separated by a large distance. A key feature is the underlying wave mediated dynamics. The particles' phase space dynamics is given by the system as a whole and cannot be described independently. Numerical phase space histograms display statistical coherence; the particles' intricate distributions in phase space are statistically indistinguishable. However, removing one particle changes the phase space picture completely, which is reminiscent of entanglement. The model here presented also relates to nonlinearly coupled oscillators where synchronization can break out spontaneously. The present oscillator-coupling is dynamic and can change intensity through the underlying wave field as opposed to, for example, the Kuramoto model where the coupling is pre-defined. There are some regimes where we observe phase-locking or, more generally, regimes where the oscillators are statistically indistinguishable in phase-space, where numerical histograms display their (mutual) most likely amplitude and phase.