Investigating how classical systems may manifest dynamics analogous to those of quantum systems is a broad subject of fundamental interest. Walking droplets, which self-propel through a resonant interaction with their own wave field, provide a unique macroscopic realization of wave-particle duality that exhibits behaviors previously thought exclusive to quantum particles. Despite significant efforts, elucidating the precise origin and form of the wave-mediated forces responsible for the walker's quantumlike behavior remained elusive. Here, we demonstrate that, owing to wave interference, the force responsible for orbital quantization originates from waves excited near stationary points on the walker's past trajectory. Moreover, we derive a minimal model with the essential ingredients to capture quantized orbital dynamics, including quasiperiodic and chaotic orbits. Notably, this minimal model provides an explicit distinction between local forces, which account for the walker's preferred speed and wave-induced added mass, and spatiotemporal nonlocal forces responsible for quantization. The quantization mechanism revealed here is generic, and will thus play a role in other hydrodynamic quantum analogs.
Investigating how classical systems may manifest dynamics analogous to those of quantum systems is a broad subject of fundamental interest. Walking droplets, which self-propel through a resonant interaction with their own wave field, provide a unique macroscopic realization of wave-particle duality that exhibits behaviors previously thought exclusive to quantum particles. Despite significant efforts, elucidating the precise origin and form of the wave-mediated forces responsible for the walker's quantumlike behavior remained elusive. Here, we demonstrate that, owing to wave interference, the force responsible for orbital quantization originates from waves excited near stationary points on the walker's past trajectory. Moreover, we derive a minimal model with the essential ingredients to capture quantized orbital dynamics, including quasiperiodic and chaotic orbits. Notably, this minimal model provides an explicit distinction between local forces, which account for the walker's preferred speed and wave-induced added mass, and spatiotemporal nonlocal forces responsible for quantization. The quantization mechanism revealed here is generic, and will thus play a role in other hydrodynamic quantum analogs.
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.
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]
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.
A droplet bouncing on a liquid bath can self-propel due to its interaction with the waves it generates. The resulting “walker” is a dynamical association where, at a macroscopic scale, a particle (the droplet) is driven by a pilot-wave field. A specificity of this system is that the wave field itself results from the superposition of the waves generated at the points of space recently visited by the particle. It thus contains a memory of the past trajectory of the particle. Here, we investigate the response of this object to forces orthogonal to its motion. We find that the resulting closed orbits present a spontaneous quantization. This is observed only when the memory of the system is long enough for the particle to interact with the wave sources distributed along the whole orbit. An additional force then limits the possible orbits to a discrete set. The wave-sustained path memory is thus demonstrated to generate a quantization of angular momentum. Because a quantum-like uncertainty was also observed recently in these systems, the nonlocality generated by path memory opens new perspectives.
Hydrodynamic quantum analogs is a nascent field initiated in 2005 by the discovery of a hydrodynamic pilot-wave system [Y. Couder, S. Protière, E. Fort, and A. Boudaoud, Nature 437, 208 (2005)]. The system consists of a millimetric droplet self-propeling along the surface of a vibrating bath through a resonant interaction with its own wave field [J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269–292 (2015)]. There are three critical ingredients for the quantum like-behavior. The first is “path memory” [A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi, and Y. Couder, J. Fluid Mech. 675, 433–463 (2011)], which renders the system non-Markovian: the instantaneous wave force acting on the droplet depends explicitly on its past. The second is the resonance condition between droplet and wave that ensures a highly structured monochromatic pilot wave field that imposes an effective potential on the walking droplet, resulting in preferred, quantized states. The third ingredient is chaos, which in several systems is characterized by unpredictable switching between unstable periodic orbits. This focus issue is devoted to recent studies of and relating to pilot-wave hydrodynamics, a field that attempts to answer the following simple but provocative question: Might deterministic chaotic pilot-wave dynamics underlie quantum statistics?
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.