Wavelike statistics from pilot-wave dynamics in a circular corral

Minimal Quantization Model in Pilot-Wave Hydrodynamics

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.

Experimental investigation of walking drops: Wave field and interaction with slit structures

While bouncing walking silicone oil droplets (walkers) do show many quantumlike phenomena, the original, most intriguing, double-slit experiment with walkers has been shown to be an overinterpretation of data. Several experiments and numerical simulations have proven that for at least some parameter region there is no randomness. Still, true randomness was claimed to be observed in an experiment on chaotically bouncing walkers. Also, most of the available phase space has not been investigated. The main goal of this paper is therefore to look for true interference and chaos in the entire phase space. Recently, we made an extensive investigation of drops interacting with slits, but still in a limited range. However, the outcome was always deterministic and only incidentally mimicked the statistics of the corresponding quantum case. We also showed that the extra interference, already seen by others, in the double-slit case was caused by reflection of waves from the outlet of the unused slit after passage and thus was not a true double-slit effect. A new theoretical treatment of bouncing drop dynamics has since given analytic relations for the associated wave field, leading to a proposal for criteria for the possible occurrence of true interference in the double-slit experiment. Satisfying these criteria, requires working close to the onset of the Faraday instability, with two spatial conditions favoring slow walkers, and a temporal condition favoring fast walkers. The regions of high velocity, where the walkers bounce periodically, and of very low velocity, with chaotically bouncing walkers, have not been comprehensively investigated so far. We have therefore looked at these regions, probing the limits for interaction with slits. Furthermore, noting that a short transit time is essential to fulfill the criteria, experiments were done using double-slit barriers only 0.5 and 2 mm broad. Nowhere was true interference or a chaotic response found. As the theory has implications for many of the observed quantumlike phenomena involving walkers as, e.g., tunneling and interaction between drops, we have measured the spatial and temporal decay of the wave field. A comparison with the theory shows very good agreement but leads to a reformulation of the above-mentioned criteria.

Probing Hydrodynamic Fluctuation-Induced Forces with an Oscillating Robot

We study the dynamics of an oscillating, free-floating robot that generates radially expanding gravity-capillary waves at a fluid surface. In open water, the device does not self-propel; near a rigid boundary, it can be attracted or repelled. Visualization of the wave field dynamics reveals that when near a boundary, a complex interference of generated and reflected waves induces a wave amplitude fluctuation asymmetry. Attraction increases as wave frequency increases or robot-boundary separation decreases. Theory on confined gravity-capillary wave radiation dynamics developed by Hocking in the 1980s captures the observed parameter dependence due to these "Hocking fields." The flexibility of the robophysical system allows detailed characterization and analysis of locally generated nonequilibrium fluctuation-induced forces [M. Kardar and R. Golestanian, Rev. Mod. Phys. 71, 1233 (1999)RMPHAT0034-686110.1103/RevModPhys.71.1233].

Infinite-memory classical wave-particle entities, attractor-driven active particles, and the diffusionless Lorenz equations

A classical wave-particle entity (WPE) can materialize as a millimeter-sized droplet walking horizontally on the free surface of a vertically vibrating liquid bath. This WPE comprises a particle (droplet) that shapes its environment by locally exciting decaying standing waves, which, in turn, guides the particle motion. At high amplitude of bath vibrations, the particle-generated waves decay very slowly in time and the particle motion is influenced by the history of waves along its trajectory. In this high-memory regime, WPEs exhibit hydrodynamic quantum analogs where quantum-like statistics arise from underlying chaotic dynamics. Exploration of WPE dynamics in the very high-memory regime requires solving an integrodifferential equation of motion. By using an idealized one-dimensional WPE model where the particle generates sinusoidal waves, we show that in the limit of infinite memory, the system dynamics reduce to a 3D nonlinear system of ordinary differential equations (ODEs) known as the diffusionless Lorenz equations (DLEs). We use our algebraically simple ODE system to explore in detail, theoretically and numerically, the rich set of periodic and chaotic dynamical behaviors exhibited by the WPE in the parameter space. Specifically, we link the geometry and dynamics in the phase-space of the DLE system to the dynamical and statistical features of WPE motion, paving a way to understand hydrodynamic quantum analogs using phase-space attractors. Our system also provides an alternate interpretation of an attractor-driven particle, i.e., an active particle driven by internal state-space variables of the DLE system. Hence, our results might also provide new insights into modeling active particle locomotion.

Pilot-wave dynamics: Using dynamic mode decomposition to characterize bifurcations, routes to chaos, and emergent 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.

Damped-driven system of bouncing droplets leading to deterministic diffusive behavior

Damped-driven systems are ubiquitous in science, however, the damping and driving mechanisms are often quite convoluted. This paper presents an experimental and theoretical investigation of a fluidic droplet on a vertically vibrating fluid bath as a damped-driven system. We study a fluidic droplet in an annular cavity with the fluid bath forced above the Faraday wave threshold. We model the droplet as a kinematic point particle in air and as inelastic collisions during impact with the bath. In both experiments and the model, the droplet is observed to chaotically change velocity with a Gaussian distribution. Finally, the statistical distributions from experiments and theory are analyzed. Incredibly, this simple deterministic interaction of damping and driving of the droplet leads to more complex Brownian-like and Levy-like behavior.

Statistical self-organization of an assembly of interacting walking drops in a confining potential

A drop bouncing on a vertically vibrated surface may self-propel forward by standing waves and travels along a fluid interface. This system called walking drop forms a non-quantum wave-particle association at the macroscopic scale. The dynamics of one particle has triggered many investigations and has resulted in spectacular experimental results in the last decade. We investigate numerically the dynamics of an assembly of walkers, i.e., a large number of walking drops evolving on a unbounded fluid interface in the presence of a confining potential acting on the particles. We show that even if the individual trajectories are erratic, the system presents a well-defined ordered internal structure that remains invariant to parameter variations such as the number of drops, the memory time and the bath radius. We rationalize such non-stationary self-organization in terms of the symmetry of the waves and show that oscillatory pair potentials form a wavy collective state of active matter.

Dynamics, interference effects, and multistability in a Lorenz-like system of a classical wave-particle entity in a periodic potential

A classical wave-particle entity (WPE) can be realized experimentally as a droplet walking on the free surface of a vertically vibrating liquid bath, with the droplet's horizontal walking motion guided by its self-generated wave field. These self-propelled WPEs have been shown to exhibit analogs of several quantum and optical phenomena. Using an idealized theoretical model that takes the form of a Lorenz-like system, we theoretically and numerically explore the dynamics of such a one-dimensional WPE in a sinusoidal potential. We find steady states of the system that correspond to a stationary WPE as well as a rich array of unsteady motions, such as back-and-forth oscillating walkers, runaway oscillating walkers, and various types of irregular walkers. In the parameter space formed by the dimensionless parameters of the applied sinusoidal potential, we observe patterns of alternating unsteady behaviors suggesting interference effects. Additionally, in certain regions of the parameter space, we also identify multistability in the particle's long-term behavior that depends on the initial conditions. We make analogies between the identified behaviors in the WPE system and Bragg's reflection of light as well as electron motion in crystals.

Superradiant Droplet Emission from Parametrically Excited Cavities

Superradiance occurs when a collection of atoms exhibits a cooperative, spontaneous emission of photons at a rate that exceeds that of its component parts. Here, we reveal a similar phenomenon in a hydrodynamic system consisting of a pair of vibrationally excited cavities, coupled through their common wave field, that spontaneously emit droplets via interfacial fracture. We show that the droplet emission rate of two coupled cavities is higher than the emission rate of two isolated cavities. Moreover, the amplified emission rate varies sinusoidally with distance between the cavities, as is characteristic of superradiance. We thus present a hydrodynamic phenomenon that captures several essential features of superradiance in optical systems.

Attractor-driven matter

The state of a classical point-particle system may often be specified by giving the position and momentum for each constituent particle. For non-pointlike particles, the center-of-mass position may be augmented by an additional coordinate that specifies the internal state of each particle. The internal state space is typically topologically simple, in the sense that the particle's internal coordinate belongs to a suitable symmetry group. In this paper, we explore the idea of giving internal complexity to the particles, by attributing to each particle an internal state space that is represented by a point on a strange (or otherwise) attracting set. It is, of course, very well known that strange attractors arise in a variety of nonlinear dynamical systems. However, rather than considering strange attractors as emerging from complex dynamics, we may employ strange attractors to drive such dynamics. In particular, by using an attractor (strange or otherwise) to model each particle's internal state space, we present a class of matter coined "attractor-driven matter." We outline the general formalism for attractor-driven matter and explore several specific examples, some of which are reminiscent of active matter. Beyond the examples studied in this paper, our formalism for attractor-driven dynamics may be applicable more broadly, to model complex dynamical and emergent behaviors in a variety of contexts.

Pseudolaminar chaos from on-off intermittency

In finite-dimensional, chaotic, Lorenz-like wave-particle dynamical systems one can find diffusive trajectories, which share their appearance with that of laminar chaotic diffusion [Phys. Rev. Lett. 128, 074101 (2022)0031-900710.1103/PhysRevLett.128.074101] known from delay systems with lag-time modulation. Applying, however, to such systems a test for laminar chaos, as proposed in [Phys. Rev. E 101, 032213 (2020)2470-004510.1103/PhysRevE.101.032213], these signals fail such a test, thus leading to the notion of pseudolaminar chaos. The latter can be interpreted as integrated periodically driven on-off intermittency. We demonstrate that, on a signal level, true laminar and pseudolaminar chaos are hardly distinguishable in systems with and without dynamical noise. However, very pronounced differences become apparent when correlations of signals and increments are considered. We compare and contrast these properties of pseudolaminar chaos with true laminar chaos.

Crises and chaotic scattering in hydrodynamic pilot-wave experiments

Theoretical foundations of chaos have been predominantly laid out for finite-dimensional dynamical systems, such as the three-body problem in classical mechanics and the Lorenz model in dissipative systems. In contrast, many real-world chaotic phenomena, e.g., weather, arise in systems with many (formally infinite) degrees of freedom, which limits direct quantitative analysis of such systems using chaos theory. In the present work, we demonstrate that the hydrodynamic pilot-wave systems offer a bridge between low- and high-dimensional chaotic phenomena by allowing for a systematic study of how the former connects to the latter. Specifically, we present experimental results, which show the formation of low-dimensional chaotic attractors upon destabilization of regular dynamics and a final transition to high-dimensional chaos via the merging of distinct chaotic regions through a crisis bifurcation. Moreover, we show that the post-crisis dynamics of the system can be rationalized as consecutive scatterings from the nonattracting chaotic sets with lifetimes following exponential distributions.

Overload wave-memory induces amnesia of a self-propelled particle

Information storage is a key element of autonomous, out-of-equilibrium dynamics, especially for biological and synthetic active matter. In synthetic active matter however, the implementation of internal memory in self-propelled systems is often absent, limiting our understanding of memory-driven dynamics. Recently, a system comprised of a droplet generating its guiding wavefield appeared as a prime candidate for such investigations. Indeed, the wavefield, propelling the droplet, encodes information about the droplet trajectory and the amount of information can be controlled by a single scalar experimental parameter. In this work, we show numerically and experimentally that the accumulation of information in the wavefield induces the loss of time correlations, where the dynamics can then be described by a memory-less process. We rationalize the resulting statistical behavior by defining an effective temperature for the particle dynamics where the wavefield acts as a thermostat of large dimensions, and by evidencing a minimization principle of the generated wavefield.

Memory and information storage play an important role in biological systems, however challenging to implement in synthetic active matter. The authors show that the wave field, propelling the particle, acts as a memory repository, and an excess of memory leads to a memory-less particle dynamics.

A classical analog of the quantum Zeeman effect

We extend a recent classical mechanical analog of Bohr's atom consisting of a scalar field coupled to a massive point-like particle [P. Jamet and A. Drezet, "A mechanical analog of Bohr's atom based on de Broglie's double-solution approach," Chaos 31, 103120 (2021)] by adding and studying the contribution of a uniform weak magnetic field on their dynamics. In doing so, we are able to recover the splitting of the energy levels of the atom called Zeeman's effect within the constraints of our model and in agreement with the semiclassical theory of Sommerfeld. This result is obtained using Larmor's theorem for both the field and the particle, associating magnetic effects with inertial Coriolis forces in a rotating frame of reference. Our work, based on the old "double solution" theory of de Broglie, shows that a dualistic model involving a particle guided by a scalar field can reproduce the normal Zeeman effect.

Lorenz-like systems emerging from an integro-differential trajectory equation of a one-dimensional wave-particle entity

Vertically vibrating a liquid bath can give rise to a self-propelled wave-particle entity on its free surface. The horizontal walking dynamics of this wave-particle entity can be described adequately by an integro-differential trajectory equation. By transforming this integro-differential equation of motion for a one-dimensional wave-particle entity into a system of ordinary differential equations (ODEs), we show the emergence of Lorenz-like dynamical systems for various spatial wave forms of the entity. Specifically, we present and give examples of Lorenz-like dynamical systems that emerge when the wave form gradient is (i) a solution of a linear homogeneous constant coefficient ODE, (ii) a polynomial, and (iii) a periodic function. Understanding the dynamics of the wave-particle entity in terms of Lorenz-like systems may prove to be useful in rationalizing emergent statistical behavior from underlying chaotic dynamics in hydrodynamic quantum analogs of walking droplets. Moreover, the results presented here provide an alternative physical interpretation of various Lorenz-like dynamical systems in terms of the walking dynamics of a wave-particle entity.

Anomalous transport of a classical wave-particle entity in a tilted potential

A classical wave-particle entity in the form of a millimetric walking droplet can emerge on the free surface of a vertically vibrating liquid bath. Such wave-particle entities have been shown to exhibit hydrodynamic analogs of quantum systems. Using an idealized theoretical model of this wave-particle entity in a tilted potential, we explore its transport behavior. The integro-differential equation of motion governing the dynamics of the wave-particle entity transforms to a Lorenz-like system of ordinary differential equations that drives the particle's velocity. Several anomalous transport regimes such as absolute negative mobility, differential negative mobility, and lock-in regions corresponding to force-independent mobility are observed. These observations motivate experiments in the hydrodynamic walking-droplet system for the experimental realizations of anomalous transport phenomena.

A mechanical analog of Bohr's atom based on de Broglie's double-solution approach

Motivated by recent developments of hydrodynamical quantum mechanical analogs [J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269-292 (2015)], we provide a relativistic model for a classical particle coupled to a scalar wave field through a holonomic constraint. In the presence of an external Coulomb field, we define a regime where the particle is guided by the wave in a way similar to the old de Broglie phase-wave proposal. Moreover, this dualistic mechanical analog of the quantum theory is reminiscent of the double-solution approach suggested by de Broglie in 1927 and is able to reproduce the Bohr-Sommerfeld semiclassical quantization formula for an electron moving in an atom.

Unsteady dynamics of a classical particle-wave entity

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.

Stop-and-go locomotion of superwalking droplets

Vertically vibrating a liquid bath at two frequencies, f and f/2, having a constant relative phase difference can give rise to self-propelled superwalking droplets on the liquid surface. We have numerically investigated such superwalking droplets in the regime where the phase difference varies slowly with time. We predict the emergence of stop-and-go motion of droplets, consistent with experimental observations [Valani et al. Phys. Rev. Lett. 123, 024503 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.024503]. Our simulations in the parameter space spanned by the droplet size and the rate of traversal of the phase difference uncover three different types of droplet motion: back-and-forth, forth-and-forth, and irregular stop-and-go motion, which we explore in detail. Our findings lay a foundation for further studies of dynamically driven droplets, whereby the droplet's motion may be guided by engineering arbitrary time-dependent phase difference functions.

Classical pilot-wave dynamics: The free particle

We present the results of a theoretical investigation into the dynamics of a vibrating particle propelled by its self-induced wave field. Inspired by the hydrodynamic pilot-wave system discovered by Yves Couder and Emmanuel Fort, the idealized pilot-wave system considered here consists of a particle guided by the slope of its quasi-monochromatic "pilot" wave, which encodes the history of the particle motion. We characterize this idealized pilot-wave system in terms of two dimensionless groups that prescribe the relative importance of particle inertia, drag and wave forcing. Prior work has delineated regimes in which self-propulsion of the free particle leads to steady or oscillatory rectilinear motion; it has further revealed parameter regimes in which the particle executes a stable circular orbit, confined by its pilot wave. We here report a number of new dynamical states in which the free particle executes self-induced wobbling and precessing orbital motion. We also explore the statistics of the chaotic regime arising when the time scale of the wave decay is long relative to that of particle motion and characterize the diffusive and rotational nature of the resultant particle dynamics. We thus present a detailed characterization of free-particle motion in this rich two-parameter family of dynamical systems.

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.

Mechanical analog of quantum bradyons and tachyons

We present a mechanical analog of a quantum wave-particle duality: a vibrating string threaded through a freely moving bead or "masslet." For small string amplitudes, the particle movement is governed by a set of nonlinear dynamical equations that couple the wave field to the masslet dynamics. Under specific conditions, the particle achieves a regime of transparency in which the field and the particle's dynamics appear decoupled. In that special case, the particle conserves its momentum and a guiding wave obeying a Klein-Gordon equation, with real or imaginary mass, emerges. Similar to the double-solution theory of de Broglie, this guiding wave is locked in phase with a modulating group wave comoving with the particle. Interestingly, both subsonic and supersonic particles can fall into a quantum regime as is the case with the slower-than-light bradyons and hypothetical, faster-than-light tachyons of particle physics.

Bifurcations and chaos in a Lorenz-like pilot-wave system

A millimetric droplet may bounce and self-propel on the surface of a vertically vibrating fluid bath, guided by its self-generated wave field. This hydrodynamic pilot-wave system exhibits a vast range of dynamics, including behavior previously thought to be exclusive to the quantum realm. We present the results of a theoretical investigation of an idealized pilot-wave model, in which a particle is guided by a one-dimensional wave that is equipped with the salient features of the hydrodynamic system. The evolution of this reduced pilot-wave system may be simplified by projecting onto a three-dimensional dynamical system describing the evolution of the particle velocity, the local wave amplitude, and the local wave slope. As the resultant dynamical system is remarkably similar in form to the Lorenz system, we utilize established properties of the Lorenz equations as a guide for identifying and elucidating several pilot-wave phenomena, including the onset and characterization of chaos.

Speed oscillations in classical pilot-wave 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.

Predictability in a hydrodynamic pilot-wave system: Resolution of walker tunneling

A walker is a macroscopic coupling of a droplet and a capillary wave field that exhibits several quantumlike properties. In 2009, Eddi et al. [Phys. Rev. Lett. 102, 240401 (2009)PRLTAO0031-900710.1103/PhysRevLett.102.240401] showed that walkers may cross a submerged barrier in an unpredictable manner and named this behavior "unpredictable walker tunneling." In quantum mechanics, tunneling is one of the simplest arrangements where similar unpredictability occurs. In this paper, we investigate how unpredictability can be unveiled for walkers through an experimental study of walker tunneling with precision. We refine both time and position measurements to take into account the fast bouncing dynamics of the system. Tunneling is shown to be unpredictable until a distance of 2.6 mm from the barrier center, where we observe the separation of reflected and transmitted trajectories in the position-velocity phase-space. The unpredictability is unlikely to be attributable to either uncertainty in the initial conditions or to the noise in the experiment. It is more likely due to changes in the drop's vertical dynamics arising when it interacts with the barrier. We compare this macroscopic system to a tunneling quantum particle that is subjected to repeated measurements of its position and momentum. We show that, despite the different theoretical treatments of these two disparate systems, similar patterns emerge in the position-velocity phase space.

A hydrodynamic analog of Friedel oscillations

We present a macroscopic analog of an open quantum system, achieved with a classical pilot-wave system. Friedel oscillations are the angstrom-scale statistical signature of an impurity on a metal surface, concentric circular modulations in the probability density function of the surrounding electron sea. We consider a millimetric drop, propelled by its own wave field along the surface of a vibrating liquid bath, interacting with a submerged circular well. An ensemble of drop trajectories displays a statistical signature in the vicinity of the well that is strikingly similar to Friedel oscillations. The droplet trajectories reveal the dynamical roots of the emergent statistics. Our study elucidates a new mechanism for emergent quantum-like statistics in pilot-wave hydrodynamics and so suggests new directions for the nascent field of hydrodynamic quantum analogs.

Superwalking Droplets

A walker is a droplet of liquid that self-propels on the free surface of an oscillating bath of the same liquid through feedback between the droplet and its wave field. We have studied walking droplets in the presence of two driving frequencies and have observed a new class of walking droplets, which we coin superwalkers. Superwalkers may be more than double the size of the largest walkers, may travel at more than triple the speed of the fastest ones, and enable a plethora of novel multidroplet behaviors.

Bouncing Oil Droplets, de Broglie's Quantum Thermostat, and Convergence to Equilibrium

Recently, the properties of bouncing oil droplets, also known as “walkers,” have attracted much attention because they are thought to offer a gateway to a better understanding of quantum behavior. They indeed constitute a macroscopic realization of wave-particle duality, in the sense that their trajectories are guided by a self-generated surrounding wave. The aim of this paper is to try to describe walker phenomenology in terms of de Broglie–Bohm dynamics and of a stochastic version thereof. In particular, we first study how a stochastic modification of the de Broglie pilot-wave theory, à la Nelson, affects the process of relaxation to quantum equilibrium, and we prove an H-theorem for the relaxation to quantum equilibrium under Nelson-type dynamics. We then compare the onset of equilibrium in the stochastic and the de Broglie–Bohm approaches and we propose some simple experiments by which one can test the applicability of our theory to the context of bouncing oil droplets. Finally, we compare our theory to actual observations of walker behavior in a 2D harmonic potential well.

Dynamics, emergent statistics, and the mean-pilot-wave potential of walking droplets

A millimetric droplet may bounce and self-propel on the surface of a vertically vibrating bath, where its horizontal "walking" motion is induced by repeated impacts with its accompanying Faraday wave field. For ergodic long-time dynamics, we derive the relationship between the droplet's stationary statistical distribution and its mean wave field in a very general setting. We then focus on the case of a droplet subjected to a harmonic potential with its motion confined to a line. By analyzing the system's periodic states, we reveal a number of dynamical regimes, including those characterized by stationary bouncing droplets trapped by the harmonic potential, periodic quantized oscillations, chaotic motion and wavelike statistics, and periodic wave-trapped droplet motion that may persist even in the absence of a central force. We demonstrate that as the vibrational forcing is increased progressively, the periodic oscillations become chaotic via the Ruelle-Takens-Newhouse route. We rationalize the role of the local pilot-wave structure on the resulting droplet motion, which is akin to a random walk. We characterize the emergence of wavelike statistics influenced by the effective potential that is induced by the mean Faraday wave field.

Walking droplets in a circular corral: Quantisation and chaos

A millimetric liquid droplet may walk across the surface of a vibrating liquid bath through a resonant interaction with its self-generated wavefield. Such walking droplets, or "walkers," have attracted considerable recent interest because they exhibit certain features previously believed to be exclusive to the microscopic, quantum realm. In particular, the intricate motion of a walker confined to a closed geometry is known to give rise to a coherent wave-like statistical behavior similar to that of electrons confined to quantum corrals. Here, we examine experimentally the dynamics of a walker inside a circular corral. We first illustrate the emergence of a variety of stable dynamical states for relatively low vibrational accelerations, which lead to a double quantisation in angular momentum and orbital radius. We then characterise the system's transition to chaos for increasing vibrational acceleration and illustrate the resulting breakdown of the double quantisation. Finally, we discuss the similarities and differences between the dynamics and statistics of a walker inside a circular corral and that of a walker subject to a simple harmonic potential.

Interaction of two walkers: Perturbed vertical dynamics as a source of chaos

Walkers are dual objects comprising a bouncing droplet dynamically coupled to an underlying Faraday wave at the surface of a vibrated bath. In this paper, we study the wave-mediated interaction of two walkers launched at one another, both experimentally and theoretically. Different outcomes are observed in which either the walkers scatter or they bind to each other in orbits or promenade-like motions. The outcome is highly sensitive to initial conditions, which is a signature of chaos, though the time during which perturbations are amplified is finite. The vertical bouncing dynamics, periodic for a single walker, is also strongly perturbed during the interaction, owing to the superposition of the wave contributions of each droplet. Thanks to a model based on inelastic balls coupled to the Faraday waves, we show that this perturbed vertical dynamics is the source of horizontal chaos in such a system.

Bouncing droplet dynamics above the Faraday threshold

We present the results of an experimental investigation of the dynamics of droplets bouncing on a vibrating fluid bath for forcing accelerations above the Faraday threshold. Two distinct fluid viscosity and vibrational frequency combinations (20 cS-80 Hz and 50 cS-50 Hz) are considered, and the dependence of the system behavior on drop size and vibrational acceleration is characterized. A number of new dynamical regimes are reported, including meandering, zig-zagging, erratic bouncing, coalescing, and trapped regimes. Particular attention is given to the regime in which droplets change direction erratically and exhibit a dynamics akin to Brownian motion. We demonstrate that the effective diffusivity increases with vibrational acceleration and decreases with drop size, as suggested by simple scaling arguments.

The interaction of a walking droplet and a submerged pillar: From scattering to the logarithmic spiral

Millimetric droplets may walk across the surface of a vibrating fluid bath, propelled forward by their own guiding or "pilot" wave field. We here consider the interaction of such walking droplets with a submerged circular pillar. While simple scattering events are the norm, as the waves become more pronounced, the drop departs the pillar along a path corresponding to a logarithmic spiral. The system behavior is explored both experimentally and theoretically, using a reduced numerical model in which the pillar is simply treated as a region of decreased wave speed. A trajectory equation valid in the limit of weak droplet acceleration is used to infer an effective force due to the presence of the pillar, which is found to be a lift force proportional to the product of the drop's walking speed and its instantaneous angular speed around the post. This system presents a macroscopic example of pilot-wave-mediated forces giving rise to apparent action at a distance.

Walking droplets correlated at a distance

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.

Hong-Ou-Mandel-like two-droplet correlations

We present a numerical study of two-droplet pair correlations for in-phase droplets walking on a vibrating bath. Two such walkers are launched toward a common point of intersection. As they approach, their carrier waves may overlap and the droplets have a non-zero probability of forming a two-droplet bound state. The likelihood of such pairing is quantified by measuring the probability of finding the droplets in a bound state at late times. Three generic types of two-droplet correlations are observed: promenading, orbiting, and chasing pair of walkers. For certain parameters, the droplets may become correlated for certain initial path differences and remain uncorrelated for others, while in other cases, the droplets may never produce droplet pairs. These observations pave the way for further studies of strongly correlated many-droplet behaviors in the hydrodynamical quantum analogs of bouncing and walking droplets.

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

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.

Introduction to focus issue on hydrodynamic quantum analogs

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?

A review of the theoretical modeling of walking droplets: Toward a generalized pilot-wave framework

The walking droplet system has extended the range of classical systems to include several features previously thought to be exclusive to quantum systems. We review the hierarchy of analytic models that have been developed, on the basis of various simplifying assumptions, to describe droplets walking on a vibrating fluid bath. Particular attention is given to detailing their successes and failures in various settings. Finally, we present a theoretical model that may be adopted to explore a more generalized pilot-wave framework capable of further extending the phenomenological range of classical pilot-wave systems beyond that achievable in the laboratory.

Hydrodynamic spin states

We present the results of a theoretical investigation of hydrodynamic spin states, wherein a droplet walking on a vertically vibrating fluid bath executes orbital motion despite the absence of an applied external field. In this regime, the walker's self-generated wave force is sufficiently strong to confine the walker to a circular orbit. We use an integro-differential trajectory equation for the droplet's horizontal motion to specify the parameter regimes for which the innermost spin state can be stabilized. Stable spin states are shown to exhibit an analog of the Zeeman effect from quantum mechanics when they are placed in a rotating frame.

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

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.

Transition to chaos in wave memory dynamics in a harmonic well: Deterministic and noise-driven behavior

A walker is the association of a sub-millimetric bouncing drop moving along with a co-evolving Faraday wave. When confined in a harmonic potential, its stable trajectories are periodic and quantised both in extension and mean angular momentum. In this article, we present the rest of the story, specifically the chaotic paths. They are chaotic and show intermittent behaviors between an unstable quantised set of attractors. First, we present the two possible situations we find experimentally. Then, we emphasise theoretically two mechanisms that lead to unstable situations. It corresponds either to noise-driven chaos or low-dimensional deterministic chaos. Finally, we characterise experimentally each of these distinct situations. This article aims at presenting a comprehensive investigation of the unstable paths in order to complete the picture of walkers in a two dimensional harmonic potential.

Atom-Diffraction from Surfaces with Defects: A Fermatian, Newtonian and Bohmian Joint View

Bohmian mechanics, widely known within the field of the quantum foundations, has been a quite useful resource for computational and interpretive purposes in a wide variety of practical problems. Here, it is used to establish a comparative analysis at different levels of approximation in the problem of the diffraction of helium atoms from a substrate consisting of a defect with axial symmetry on top of a flat surface. The motivation behind this work is to determine which aspects of one level survive in the next level of refinement and, therefore, to get a better idea of what we usually denote as quantum-classical correspondence. To this end, first a quantum treatment of the problem is performed with both an approximated hard-wall model and then with a realistic interaction potential model. The interpretation and explanation of the features displayed by the corresponding diffraction intensity patterns is then revisited with a series of trajectory-based approaches: Fermatian trajectories (optical rays), Newtonian trajectories and Bohmian trajectories. As it is seen, while Fermatian and Newtonian trajectories show some similarities, Bohmian trajectories behave quite differently due to their implicit non-classicality.

Two-frequency forcing of droplet rebounds on a liquid bath

Droplets can bounce indefinitely on a liquid bath vertically vibrated in a sinusoidal fashion. We here present experimental results that extend this observation to forcing signals composed of a combination of two commensurable frequencies. The Faraday and Goodridge thresholds are characterized. Then a number of vertical bouncing modes are reported, including walkers. The vertical motion can become chaotic, in which case the horizontal motion is an alternation of walk and stop.

Self-attraction into spinning eigenstates of a mobile wave source by its emission back-reaction

The back-reaction of a radiated wave on the emitting source is a general problem. In the most general case, back-reaction on moving wave sources depends on their whole history. Here we study a model system in which a pointlike source is piloted by its own memory-endowed wave field. Such a situation is implemented experimentally using a self-propelled droplet bouncing on a vertically vibrated liquid bath and driven by the waves it generates along its trajectory. The droplet and its associated wave field form an entity having an intrinsic dual particle-wave character. The wave field encodes in its interference structure the past trajectory of the droplet. In the present article we show that this object can self-organize into a spinning state in which the droplet possesses an orbiting motion without any external interaction. The rotation is driven by the wave-mediated attractive interaction of the droplet with its own past. The resulting "memory force" is investigated and characterized experimentally, numerically, and theoretically. Orbiting with a radius of curvature close to half a wavelength is shown to be a memory-induced dynamical attractor for the droplet's motion.

The onset of chaos in orbital pilot-wave dynamics

We present the results of a numerical investigation of the emergence of chaos in the orbital dynamics of droplets walking on a vertically vibrating fluid bath and acted upon by one of the three different external forces, specifically, Coriolis, Coulomb, or linear spring forces. As the vibrational forcing of the bath is increased progressively, circular orbits destabilize into wobbling orbits and eventually chaotic trajectories. We demonstrate that the route to chaos depends on the form of the external force. When acted upon by Coriolis or Coulomb forces, the droplet's orbital motion becomes chaotic through a period-doubling cascade. In the presence of a central harmonic potential, the transition to chaos follows a path reminiscent of the Ruelle-Takens-Newhouse scenario.

Quantumlike statistics of deterministic wave-particle interactions in a circular cavity

A deterministic low-dimensional iterated map is proposed here to describe the interaction between a bouncing droplet and Faraday waves confined to a circular cavity. Its solutions are investigated theoretically and numerically. The horizontal trajectory of the droplet can be chaotic: it then corresponds to a random walk of average step size equal to half the Faraday wavelength. An analogy is made between the diffusion coefficient of this random walk and the action per unit mass ℏ/m of a quantum particle. The statistics of droplet position and speed are shaped by the cavity eigenmodes, in remarkable agreement with the solution of Schrödinger equation for a quantum particle in a similar potential well.

Pilot-wave dynamics in a harmonic potential: Quantization and stability of circular orbits

We present the results of a theoretical investigation of the dynamics of a droplet walking on a vibrating fluid bath under the influence of a harmonic potential. The walking droplet's horizontal motion is described by an integro-differential trajectory equation, which is found to admit steady orbital solutions. Predictions for the dependence of the orbital radius and frequency on the strength of the radial harmonic force field agree favorably with experimental data. The orbital quantization is rationalized through an analysis of the orbital solutions. The predicted dependence of the orbital stability on system parameters is compared with experimental data and the limitations of the model are discussed.

Double-slit experiment with single wave-driven particles and its relation to quantum mechanics

In a thought-provoking paper, Couder and Fort [Phys. Rev. Lett. 97, 154101 (2006)] describe a version of the famous double-slit experiment performed with droplets bouncing on a vertically vibrated fluid surface. In the experiment, an interference pattern in the single-particle statistics is found even though it is possible to determine unambiguously which slit the walking droplet passes. Here we argue, however, that the single-particle statistics in such an experiment will be fundamentally different from the single-particle statistics of quantum mechanics. Quantum mechanical interference takes place between different classical paths with precise amplitude and phase relations. In the double-slit experiment with walking droplets, these relations are lost since one of the paths is singled out by the droplet. To support our conclusions, we have carried out our own double-slit experiment, and our results, in particular the long and variable slit passage times of the droplets, cast strong doubt on the feasibility of the interference claimed by Couder and Fort. To understand theoretically the limitations of wave-driven particle systems as analogs to quantum mechanics, we introduce a Schrödinger equation with a source term originating from a localized particle that generates a wave while being simultaneously guided by it. We show that the ensuing particle-wave dynamics can capture some characteristics of quantum mechanics such as orbital quantization. However, the particle-wave dynamics can not reproduce quantum mechanics in general, and we show that the single-particle statistics for our model in a double-slit experiment with an additional splitter plate differs qualitatively from that of quantum mechanics.

Interaction of two walkers: wave-mediated energy and force

A bouncing droplet, self-propelled by its interaction with the waves it generates, forms a classical wave-particle association called a "walker." Previous works have demonstrated that the dynamics of a single walker is driven by its global surface wave field that retains information on its past trajectory. Here we investigate the energy stored in this wave field for two coupled walkers and how it conveys an interaction between them. For this purpose, we characterize experimentally the "promenade modes" where two walkers are bound and propagate together. Their possible binding distances take discrete values, and the velocity of the pair depends on their mutual binding. The mean parallel motion can be either rectilinear or oscillating. The experimental results are recovered analytically with a simple theoretical framework. A relation between the kinetic energy of the droplets and the total energy of the standing waves is established.

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

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.