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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Classical pilot-wave dynamics in three dimensions

Adam Forrest Kay, Matthew Durey, John W. M. Bush

Published Aug 1, 2025• 1 citation
Publisher

Abstract

Inspired by the hydrodynamic pilot-wave system and its successes as a basis for hydrodynamic quantum analogues, we present the results of a theoretical investigation of classical pilot-wave dynamics in three dimensions. We consider a particle emitting a spherically symmetric monochromatic wave, then moving in response to the local wave gradient. We demonstrate that, beyond a critical coupling constant, the stationary state destabilizes into a self-propelling state. We compute the quasi-monochromatic form of the accompanying pilot-wave field, and its dependence on the relevant control parameters, specifically the particle inertia and longevity of the pilot-wave field. The similarities between the dynamics arising in the three-dimensional system and its two-dimensional counterpart suggest that many of the quantum features captured in two dimensions will be robust to the transition to three dimensions. We discover a family of dynamical states peculiar to the three-dimensional geometry considered, including helical spin states in which the particle velocity has both angular and rectilinear components. These helical spin states are found to be quantized in both radius of curvature and pitch length and illustrate how classical pilot-wave dynamics may support a feature reminiscent of quantum spin. More exotic dynamical states are also explored and detailed.

DOI
10.1098/rspa.2024.0986

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Inspired by the hydrodynamic pilot-wave system and its successes as a basis for hydrodynamic quantum analogues, we present the results of a theoretical investigation of classical pilot-wave dynamics in three dimensions. We consider a particle emitting a spherically symmetric monochromatic wave, then moving in response to the local wave gradient. We demonstrate that, beyond a critical coupling constant, the stationary state destabilizes into a self-propelling state. We compute the quasi-monochromatic form of the accompanying pilot-wave field, and its dependence on the relevant control parameters, specifically the particle inertia and longevity of the pilot-wave field. The similarities between the dynamics arising in the three-dimensional system and its two-dimensional counterpart suggest that many of the quantum features captured in two dimensions will be robust to the transition to three dimensions. We discover a family of dynamical states peculiar to the three-dimensional geometry considered, including helical spin states in which the particle velocity has both angular and rectilinear components. These helical spin states are found to be quantized in both radius of curvature and pitch length and illustrate how classical pilot-wave dynamics may support a feature reminiscent of quantum spin. More exotic dynamical states are also explored and detailed.
We consider the hydrodynamic quantum field theory proposed by Dagan and Bush, a model of quantum dynamics inspired by Louis de Broglie and informed by the hydrodynamic pilot-wave system discovered by Couder and Fort. According to this theory, a quantum particle has an internal vibration at twice the Compton frequency that generates disturbances in an ambient scalar field, the result being self-propulsion of the particle through a resonant interaction with its pilot-wave field. Particular attention is given here to providing theoretical rationale for the geometric form of the wave field generated by steady, rectilinear particle motion at a prescribed speed, where signatures of both the de Broglie and Compton wavelengths are generally evident. While focus is given to the one-dimensional geometry considered by Dagan and Bush, we also deduce the form of the pilot wave in two dimensions. We further consider the influence on the pilot-wave form of the details of the particle-induced wave generation, specifically the spatial extent and vibration frequency of the particle. Finally, guided by analogous theoretical descriptions of the hydrodynamic system, we recast the particle dynamics in terms of an integro-differential trajectory equation. Analysis of this equation in the non-relativistic limit reveals a critical wave-particle coupling parameter, above which the particle self-propels. Our results provide the foundation for subsequent theoretical investigations of hydrodynamic quantum field theory, including the stability analysis of various dynamical states.
We revisit de Broglie’s double-solution pilot-wave theory in light of insights gained from the hydrodynamic pilot-wave system discovered by Couder and Fort [1]. de Broglie proposed that quantum particles are characterized by an internal oscillation at the Compton frequency, at which rest mass energy is exchanged with field energy. He further proposed that the resulting pilot-wave field satisfies the Klein–Gordon equation. While he developed a guidance equation for the particle, he did not specify how the particle generates the wave. Informed by the hydrodynamic pilot-wave system, we explore a variant of de Broglie’s mechanics in which the form of the Compton-scale dynamic interaction between particle and pilot wave is specified. The particle is modeled as a localized periodic disturbance of the Klein–Gordon field at twice the Compton frequency. We simulate the evolution of the particle position by assuming that the particle is propelled by the local gradient of its pilot wave field. Resonance is achieved between the particle and its pilot wave, leading to self-excited motion of the particle. The particle locks into quasi-steady motion characterized by a mean momentum p ¯ = ℏ k , where k is the wavenumber of the surrounding matter waves. Speed modulations along the particle path arise with the de Broglie wavelength and frequency c k . The emergent dynamics is strongly reminiscent of that arising in the hydrodynamic pilot-wave system, on the basis of which we anticipate the emergence of quantum statistics in various settings. Our results suggest the potential value of a new hydrodynamically-inspired pilot-wave theory for the motion of quantum particles.
The walking droplet system discovered by Yves Couder and Emmanuel Fort presents an example of a vibrating particle self-propelling through a resonant interaction with its own wave field. It provides a means of visualizing a particle as an excitation of a field, a common notion in quantum field theory. Moreover, it represents the first macroscopic realization of a form of dynamics proposed for quantum particles by Louis de Broglie in the 1920s. The fact that this hydrodynamic pilot-wave system exhibits many features typically associated with the microscopic, quantum realm raises a number of intriguing questions. At a minimum, it extends the range of classical systems to include quantum-like statistics in a number of settings. A more optimistic stance is that it suggests the manner in which quantum mechanics might be completed through a theoretical description of particle trajectories. We here review the experimental studies of the walker system, and the hierarchy of theoretical models developed to rationalize its behavior. Particular attention is given to enumerating the dynamical mechanisms responsible for the emergence of robust, structured statistical behavior. Another focus is demonstrating how the temporal nonlocality of the droplet dynamics, as results from the persistence of its pilot wave field, may give rise to behavior that appears to be spatially nonlocal. Finally, we describe recent explorations of a generalized theoretical framework that provides a mathematical bridge between the hydrodynamic pilot-wave system and various realist models of quantum dynamics.
We present the results of a theoretical investigation of a dynamical system consisting of a particle self-propelling through a resonant interaction with its own quasi-monochromatic pilot-wave field. We rationalize two distinct mechanisms, arising in different regions of parameter space, that may lead to a wavelike statistical signature with the pilot-wavelength. First, resonant speed oscillations with the wavelength of the guiding wave may arise when the particle is perturbed from its steady self-propelling state. Second, a random-walk-like motion may set in when the decay rate of the pilot-wave field is sufficiently small. The implications for the emergent statistics in classical pilot-wave systems are discussed.
Yves Couder, Emmanuel Fort, and coworkers recently discovered that a millimetric droplet sustained on the surface of a vibrating fluid bath may self-propel through a resonant interaction with its own wave field. This article reviews experimental evidence indicating that the walking droplets exhibit certain features previously thought to be exclusive to the microscopic, quantum realm. It then reviews theoretical descriptions of this hydrodynamic pilot-wave system that yield insight into the origins of its quantum-like behavior. Quantization arises from the dynamic constraint imposed on the droplet by its pilot-wave field, and multimodal statistics appear to be a feature of chaotic pilot-wave dynamics. I attempt to assess the potential and limitations of this hydrodynamic system as a quantum analog. This fluid system is compared to quantum pilot-wave theories, shown to be markedly different from Bohmian mechanics and more closely related to de Broglie's original conception of quantum dynamics, his double-solution theory, and its relatively recent extensions through researchers in stochastic electrodynamics.
We present a number of fresh perspectives on pilot-wave hydrodynamics, the field initiated in 2005 by Couder and Fort's discovery that millimetric droplets self-propelling along the surface of a vibrating bath can capture certain features of quantum systems. A recurring theme will be that pilot-wave hydrodynamics furnishes a classical framework for reproducing many quantum phenomena and allows one to rationalize such phenomena mechanistically, from a local realist perspective, obviating the need to appeal to quantum nonlocality. The distinction is drawn between hydrodynamic pilot-wave theory and its quantum counterparts, Bohmian mechanics, the Bohm–Vigier stochastic pilot-wave theory, and de Broglie's theory of the double-solution. Each of these quantum predecessors provide a valuable touchstone as we take the physical picture engendered in the walking droplets and extend it into the quantum realm via theoretical modeling. Emphasis is given to recent developments in the field, both experimental and conceptual, and to forecasting potentially fruitful new directions.