We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry and apply the dynamic mode decomposition (DMD) in order to characterize the evolution of the wave field as the bath's vibrational acceleration is increased progressively. Dynamic mode decomposition provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatiotemporal data. Thus, DMD reduces the complex nonlinear interactions between pilot waves and droplet to a low-dimensional linear superposition of DMD modes characterizing the wave field. In particular, it provides a low-dimensional characterization of the bifurcation structure of the pilot-wave physics, wherein the excitation and recruitment of additional modes in the linear superposition models the bifurcation sequence. This DMD characterization yields a fresh perspective on the bouncing-droplet problem that forges valuable new links with the mathematical machinery of quantum mechanics. Specifically, the analysis shows that as the vibrational acceleration is increased, the pilot-wave field undergoes a series of Hopf bifurcations that ultimately lead to a chaotic wave field. The established relation between the mean pilot-wave field and the droplet statistics allows us to characterize the evolution of the emergent statistics with increased vibrational forcing from the evolution of the pilot-wave field. We thus develop a numerical framework with the same basic structure as quantum mechanics, specifically a wave theory that predicts particle statistics.
We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry and apply the dynamic mode decomposition (DMD) in order to characterize the evolution of the wave field as the bath's vibrational acceleration is increased progressively. Dynamic mode decomposition provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatiotemporal data. Thus, DMD reduces the complex nonlinear interactions between pilot waves and droplet to a low-dimensional linear superposition of DMD modes characterizing the wave field. In particular, it provides a low-dimensional characterization of the bifurcation structure of the pilot-wave physics, wherein the excitation and recruitment of additional modes in the linear superposition models the bifurcation sequence. This DMD characterization yields a fresh perspective on the bouncing-droplet problem that forges valuable new links with the mathematical machinery of quantum mechanics. Specifically, the analysis shows that as the vibrational acceleration is increased, the pilot-wave field undergoes a series of Hopf bifurcations that ultimately lead to a chaotic wave field. The established relation between the mean pilot-wave field and the droplet statistics allows us to characterize the evolution of the emergent statistics with increased vibrational forcing from the evolution of the pilot-wave field. We thus develop a numerical framework with the same basic structure as quantum mechanics, specifically a wave theory that predicts particle statistics.
A pilot-wave hydrodynamic system consists of a small droplet of viscous fluid that is propelled across a vibrating bath of the same fluid. Bouncing vertical motion and "walking" horizontal motion of the droplet can be achieved with careful control over the frequency and amplitude of the bath's oscillation. The droplet avoids merging with the fluid from which it was generated and exhibits intricate motion across the fluid surface. Of particular importance, the observed walking motion is due to the droplet's interaction with the waves it generates as it bounces off of the vibrating liquid surface. With each rebound, it receives transverse kicks in its motion dependent on the waves generated by its previous bounces. Ultimately, it is the droplet's interaction with the superposition of these waves that produce the complex motion observed. With variations in bath geometries and subsurface structures, the droplet's trajectory can be manipulated to induce fascinating behavior. Over short timescales, a droplet will exhibit seemingly random trajectories. However, when the droplet is observed over long timescales, patterns in the cumulative motion of the droplet begin to emerge. The patterns this system maps out over long timescales represent a compelling macroscopic analog to Louis de Broglie's double-solution theory of quantum mechanics. In order to analyze this behavior, a novel imaging scheme and data acquisition system has been developed, enabling observations of droplet behavior over various timescales simultaneously. An exploration into the development of a pilot-wave hydrodynamic system, and the pilot-wave hydrodynamic experiments conducted thereafter, is presented.
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 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.
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?
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 millimetric droplet bouncing on the surface of a vibrating fluid bath can self-propel by virtue of a resonant interaction with its own wave field. This system represents the first known example of a pilot-wave system of the form envisaged by Louis de Broglie in his double-solution pilot-wave theory. We here develop a fluid model of pilot-wave hydrodynamics by coupling recent models of the droplet’s bouncing dynamics with a more realistic model of weakly viscous quasi-potential wave generation and evolution. The resulting model is the first to capture a number of features reported in experiment, including the rapid transient wave generated during impact, the Doppler effect and walker–walker interactions.