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:1806.10264 · Chaos An Interdisciplinary Journal of Nonlinear Science

Pilot-wave dynamics of two identical, in-phase bouncing droplets

Rahil N. Valani, Anja C. Slim

Published Sep 1, 2018• 15 citations
arXivHTMLPDF

Abstract

A droplet bouncing on the surface of a vibrating liquid bath can move horizontally guided by the wave it produces on impacting the bath. The wave itself is modified by the environment, and thus, the interactions of the moving droplet with the surroundings are mediated through the wave. This forms an example of a pilot-wave system. Taking the Oza–Rosales–Bush description for walking droplets as a theoretical pilot-wave model, we investigate the dynamics of two interacting identical, in-phase bouncing droplets theoretically and numerically. A remarkably rich range of behaviors is encountered as a function of the two system parameters, the ratio of inertia to drag, κ, and the ratio of wave forcing to drag, β. The droplets typically travel together in a tightly bound pair, although they unbind when the wave forcing is large and inertia is small or inertia is moderately large and wave forcing is moderately small. Bound pairs can exhibit a range of trajectories depending on parameter values, including straight lines, sub-diffusive random walks, and closed loops. The droplets themselves may maintain their relative positions, oscillate toward and away from one another, or interchange positions regularly or chaotically as they travel. We explore these regimes and others and the bifurcations between them through analytic and numerical linear stability analyses and through fully nonlinear numerical simulation.

DOI
10.1063/1.5032128

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

Nonresonant effects in pilot-wave hydrodynamics

Bauyrzhan K. Primkulov, Davis J. Evans, Joel B. Been +1 more

Pilot-wave hydrodynamics concerns the dynamics of droplets walking on a vibrating liquid bath, and forms the basis for the field of hydrodynamic quantum analogs. We here investigate a theoretical model that captures both vertical and horizontal drop dynamics. The model provides new rationale for a number of phenomena, including colinear swaying, intermittent walking, and chaotic speed oscillations, all of which are linked to variability in the droplet's impact phase. Our study also highlights the degeneracy in the droplet's vertical dynamics, consideration of which is essential for understanding the dynamics of droplets in confined geometries and interacting with standing Faraday waves.

arXiv:2411.14996•Physical Review Fluids•Jan 6, 2025•10 citations

Unsteady dynamics of a classical particle-wave entity

Rahil N. Valani, Anja C. Slim, David M. Paganin +2 more

A droplet bouncing on the surface of a vertically vibrating liquid bath can walk horizontally, guided by the waves it generates on each impact. This results in a self-propelled classical particle-wave entity. By using a one-dimensional theoretical pilot-wave model with a generalized wave form, we investigate the dynamics of this particle-wave entity. We employ different spatial wave forms to understand the role played by both wave oscillations and spatial wave decay in the walking dynamics. We observe steady walking motion as well as unsteady motions such as oscillating walking, self-trapped oscillations, and irregular walking. We explore the dynamical and statistical aspects of irregular walking and show an equivalence between the droplet dynamics and the Lorenz system, as well as making connections with the Langevin equation and deterministic diffusion.

arXiv:2012.10027•Physical review. E•Jul 8, 2021•24 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

Exploring orbital dynamics and trapping with a generalized pilot-wave framework

Lucas D. Tambasco, John W. M. Bush

We explore the effects of an imposed potential with both oscillatory and quadratic components on the dynamics of walking droplets. We first conduct an experimental investigation of droplets walking on a bath with a central circular well. The well acts as a source of Faraday waves, which may trap walking droplets on circular orbits. The observed orbits are stable and quantized, with preferred radii aligning with the extrema of the well-induced Faraday wave pattern. We use the stroboscopic model of Oza et al. [J. Fluid Mech. 737, 552–570 (2013)] with an added potential to examine the interaction of the droplet with the underlying well-induced wavefield. We show that all quantized orbits are stable for low vibrational accelerations. Smaller orbits may become unstable at higher forcing accelerations and transition to chaos through a path reminiscent of the Ruelle-Takens-Newhouse scenario. We proceed by considering a generalized pilot-wave system in which the relative magnitudes of the pilot-wave force and drop inertia may be tuned. When the drop inertia is dominated by the pilot-wave force, all circular orbits may become unstable, with the drop chaotically switching between them. In this chaotic regime, the statistically stationary probability distribution of the drop’s position reflects the relative instability of the unstable circular orbits. We compute the mean wavefield from a chaotic trajectory and confirm its predicted relationship with the particle’s probability density function.

Chaos An Interdisciplinary Journal of Nonlinear Science•Sep 1, 2018•21 citations

Transition Orbits of Walking Droplets

Joshua Parker

It was recently discovered that millimeter-sized droplets bouncing on the surface of an oscillating bath of the same fluid can couple with the surface waves it produces and begin walking across the fluid bath. These walkers have been shown to behave similarly to quantum particles; a few examples include single-particle diffraction, tunneling, and quantized orbits. Such behavior occurs because the drop and surface waves depend on each other to exist, making this the first and only known macroscopic pilot-wave system. In this paper, the quantized orbits between two identical drops are explored. By sending a perturbation to a pair of orbiting walkers, the orbit can be disrupted and transition to a new orbit. The numerical results of such transitions are analyzed and discussed.

•Jan 1, 2015•0 citations
A droplet bouncing on the surface of a vibrating liquid bath can move horizontally guided by the wave it produces on impacting the bath. The wave itself is modified by the environment, and thus, the interactions of the moving droplet with the surroundings are mediated through the wave. This forms an example of a pilot-wave system. Taking the Oza–Rosales–Bush description for walking droplets as a theoretical pilot-wave model, we investigate the dynamics of two interacting identical, in-phase bouncing droplets theoretically and numerically. A remarkably rich range of behaviors is encountered as a function of the two system parameters, the ratio of inertia to drag, κ, and the ratio of wave forcing to drag, β. The droplets typically travel together in a tightly bound pair, although they unbind when the wave forcing is large and inertia is small or inertia is moderately large and wave forcing is moderately small. Bound pairs can exhibit a range of trajectories depending on parameter values, including straight lines, sub-diffusive random walks, and closed loops. The droplets themselves may maintain their relative positions, oscillate toward and away from one another, or interchange positions regularly or chaotically as they travel. We explore these regimes and others and the bifurcations between them through analytic and numerical linear stability analyses and through fully nonlinear numerical simulation.
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 hydrodynamics concerns the dynamics of droplets walking on a vibrating liquid bath, and forms the basis for the field of hydrodynamic quantum analogs. We here investigate a theoretical model that captures both vertical and horizontal drop dynamics. The model provides new rationale for a number of phenomena, including colinear swaying, intermittent walking, and chaotic speed oscillations, all of which are linked to variability in the droplet's impact phase. Our study also highlights the degeneracy in the droplet's vertical dynamics, consideration of which is essential for understanding the dynamics of droplets in confined geometries and interacting with standing Faraday waves.
A droplet bouncing on the surface of a vertically vibrating liquid bath can walk horizontally, guided by the waves it generates on each impact. This results in a self-propelled classical particle-wave entity. By using a one-dimensional theoretical pilot-wave model with a generalized wave form, we investigate the dynamics of this particle-wave entity. We employ different spatial wave forms to understand the role played by both wave oscillations and spatial wave decay in the walking dynamics. We observe steady walking motion as well as unsteady motions such as oscillating walking, self-trapped oscillations, and irregular walking. We explore the dynamical and statistical aspects of irregular walking and show an equivalence between the droplet dynamics and the Lorenz system, as well as making connections with the Langevin equation and deterministic diffusion.
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.
We explore the effects of an imposed potential with both oscillatory and quadratic components on the dynamics of walking droplets. We first conduct an experimental investigation of droplets walking on a bath with a central circular well. The well acts as a source of Faraday waves, which may trap walking droplets on circular orbits. The observed orbits are stable and quantized, with preferred radii aligning with the extrema of the well-induced Faraday wave pattern. We use the stroboscopic model of Oza et al. [J. Fluid Mech. 737, 552–570 (2013)] with an added potential to examine the interaction of the droplet with the underlying well-induced wavefield. We show that all quantized orbits are stable for low vibrational accelerations. Smaller orbits may become unstable at higher forcing accelerations and transition to chaos through a path reminiscent of the Ruelle-Takens-Newhouse scenario. We proceed by considering a generalized pilot-wave system in which the relative magnitudes of the pilot-wave force and drop inertia may be tuned. When the drop inertia is dominated by the pilot-wave force, all circular orbits may become unstable, with the drop chaotically switching between them. In this chaotic regime, the statistically stationary probability distribution of the drop’s position reflects the relative instability of the unstable circular orbits. We compute the mean wavefield from a chaotic trajectory and confirm its predicted relationship with the particle’s probability density function.
It was recently discovered that millimeter-sized droplets bouncing on the surface of an oscillating bath of the same fluid can couple with the surface waves it produces and begin walking across the fluid bath. These walkers have been shown to behave similarly to quantum particles; a few examples include single-particle diffraction, tunneling, and quantized orbits. Such behavior occurs because the drop and surface waves depend on each other to exist, making this the first and only known macroscopic pilot-wave system. In this paper, the quantized orbits between two identical drops are explored. By sending a perturbation to a pair of orbiting walkers, the orbit can be disrupted and transition to a new orbit. The numerical results of such transitions are analyzed and discussed.