FIGURE 4

# Wave fields corresponding to different steady-state dynamics at droplet impact. The droplet is denoted by a blue circle for in-phase impacts and a red circle when in flight for antiphase dynamics. Walking droplets for (a) δx = 0.065 with Γ /Γ F = 0.9 and (b) δx = 0.08 with Γ /Γ F = 0.96. Two anticlockwise orbiting droplets with Γ /Γ F = 0.91 for (c) in-phase and (d) antiphase impacts. A single droplet orbiting anticlockwise under a central force for Γ /Γ F = 0.975 with orbit radius (e) R d = 0.45 and ( f ) R d = 0.95.

Source publication

A droplet may ‘walk’ across the surface of a vertically vibrating bath of the same fluid, due to the propulsive interaction with its wave field. This hydrodynamic pilot-wave system exhibits many dynamics previously believed to exist only in the quantum realm. Starting from first principles, we derive a discrete-time fluid model, whereby the bath–dr...

## Contexts in source publication

**Context 1**

... is formulated as many 4 × 4 linear systems subject to (6.6), akin to finding the walking states. An example in-phase wave field is given in figure 4(c). ...

**Context 2**

... wave amplitude maps (6.4), but with t n+1 → t n+1/2 and δθ → π + δθ /2. An example wave field is shown in figure 4(d). The stability analysis requires a two-stage transition map to account for the antiphase impacts. ...

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When a solid body floats at the interface of a vibrating liquid bath, the motion of the object generates outwardly propagating surface waves. We here demonstrate that chiral objects on a vibrating fluid interface are set into steady rotation, with the angular speed and direction of rotation controlled by the interplay between object geometry and dr...

## Citations

... All of these features are reminiscent of kink-antikink collisions and scattering [40,39,35], and Appendices A.1 and A.2 provide further conceptual links between the timecontinuous version of the proposed model (4) and the collective coordinate description of solitary wave interactions. In future work, it would also be of interest to determine whether more refined models of walkers, such as those in [41,42], also lead to single and multi-bounce behavior, bound states, and bounce windows with fractal structure. We expect that they would. ...

Discrete dynamical models of walking droplets (“walkers”) have allowed swift numerical experiments revealing heretofore unobserved quantum statistics and related behaviors in a classical hydrodynamic system. We present evidence that one such model of walking droplets exhibits the empirically elusive
-bounce resonances that are traditionally seen in the scattering of solitary waves governed by covariant nonlinear field theories with polynomial self-interaction. A numerical investigation of the chosen model of interacting walking droplets reveals a fractal structure of resonances in the velocity in–velocity out diagram, much like the usual maps constructed for collisions of solitary waves. We suggest avenues for further theoretical analysis of walker collisions, which may connect this discrete model to the field-theoretic setting, as well as directions towards new experimental realizations
-bounce resonances.

... All of these features are reminiscent of kink-antikink collisions and scattering [40,39,35], and Appendices A.1 and A.2 provide further conceptual links between the timecontinuous version of the proposed model (4) and the collective coordinate description of solitary wave interactions. In future work, it would also be of interest to determine whether more refined models of walkers, such as those in [41,42], also lead to single and multi-bounce behavior, bound states, and bounce windows with fractal structure. We expect that they would. ...

Discrete dynamical models of walking droplets (``walkers'') have allowed swift numerical experiments revealing heretofore unobserved quantum statistics and related behaviors in a classical hydrodynamic system. One such model shows evidence of the empirically elusive $n$-bounce resonance and chaotic scattering in solitary-wave interactions investigated using covariant nonlinear field theory with polynomial self-interaction. We present experimentally testable predictions of $n$-bounce resonances between walkers. An exhaustive numerical investigation of the model reveals the usual fractal structure of resonances in the velocity in--velocity out diagram for colliding walkers. We suggest some avenues for further theoretical analysis of walker collisions, which may connect this discrete model back to the field-theoretic setting.

... The probability distribution of the walker's radius of curvature thus converges to a peaked multimodal form in the long-time limit, a finding that was corroborated by a numerical study using the stroboscopic model [54]. Experimental [58] and numerical [40,19] studies of a walker in a harmonic potential similarly revealed that, in the long-time limit, the trajectories exhibit a quantization in their radius and angular momentum. Studies of a walker in circular [34,29,18,60] and elliptical [62] "corral" geometries have shown that the long-time statistical behavior of the walker's position is related to the eigenmodes of the domain. ...

... It is evident that, after a sufficiently long time, the droplet's probability distribution converges to a multimodal form with peaks at specific radii. As described in Section 1, qualitatively similar behavior has been reported in experiments and numerical simulations of droplets in the presence of a Coriolis force [33,54], harmonic potential [58,40,19], obstacle [63] and confinement [34,29,18,62]. We leave the detailed characterization of the invariant measure of (1.1) for future work. ...

We study the long time statistics of a walker in a hydrodynamic pilot-wave system, which is a stochastic Langevin dynamics with an external potential and memory kernel. While prior experiments and numerical simulations have indicated that the system may reach a statistically steady state, its long-time behavior has not been studied rigorously. For a broad class of nonlinearities, we construct the solutions as a dynamics evolving on suitable path spaces. Then, under the assumption that the pilot-wave force is dominated by the potential, we demonstrate that the walker possesses a unique statistical steady state. We conclude by presenting an example of such an invariant measure, as obtained from a numerical simulation of a walker in a harmonic potential.

... Inspired by Couder and Fort (4), these types of bouncing droplet experiments, known as Hydrodynamic Quantum Analogs (HQAs), have been shown to produce many emergent behaviors analogous to behaviors that were previously thought to be exclusively quantum, such as quantum tunneling (5, 6, 7), Landau levels (8,9), Zeeman splitting (10), Friedell oscillations (11,12), Quantized orbits in a rotating frame (8,9,13) or a harmonic potential (14,15,16), quantum-like statistics in orbital stability (9,13), effects similar to quantum superposition and the quantum mirage (17), and much more. ...

It is known from quantum mechanics that particles are associated with wave functions whose time evolution is governed by a wave equation, and that the probability of observing the particle at some future location is proportional to the amplitude of the wave function. Although this statistical relationship is well quantified, the interpretations have remained controversial, with many split between the classical Copenhagen interpretation and some variation of de Broglie-Bohmian pilot wave models. Recent experiments with Hydrodynamic Quantum Analogs have demonstrated that many quantum effects can be modeled as emergent phenomenon from fluid dynamics, suggesting there may be a potential underlying fluid dynamics explanations for the de Broglie-Bohm pilot wave models. However, no theory has ever been presented that successfully unifies these quantum mechanical observations with the cosmic scale observations of general relativity and inflation. In this paper, we propose a novel Quantum Fluid Dynamics model of quantum mechanics, from which the concepts of particles, waves, rest mass, inertia, momentum, general relativity, mass-energy equivalence, the uncertainty principle and the appearance of a time-like dimension can all be described as emergent phenomenon from fluid dynamics.

... Experimentally, a droplet bounces periodically on a vertically and sinusoidally oscillating oil surface. The result is the emergence of a complex standing wavefield 15,16 which propels the droplet and stores information about its past positions [17][18][19][20][21][22][23][24][25][26] . Indeed, each bounce of the droplet imprints a standing longlasting axisymmetric wave centered at the point of impact, which lifetime is controlled by the vertical acceleration magnitude of the bath. ...

... Walker as a memory-driven agent. A walker is the association of a sub-millimetric oil droplet, periodically bouncing on a vertically-vibrated oil surface, and the guiding standing waves generating by the drop bounces (see Fig. 1a) [17][18][19][20][21][22][23][24][25][26] (see Methods). In this article, for practical reasons, the space available to the walker is bounded by confining the particle with an applied external potential. ...

... In Eq. (2), the memory parameter Me acts as a control parameter which allows to numerically span several and different dynamical regimes (Fig. 1d, e and f, see also Fig 1d), a quantized set of close-looped trajectories are observed and result from a wave-energy minimization 33,53 . As the memory is further increased in this intermediate regime (Fig 1e), an intermittent and chaotic dynamics is triggered where the trajectory navigates between the many possible eigenmodes of the dynamics, as investigated experimentally in 32 and theoretically in 26,42 . While a chaotic behavior has been measured and characterized in this regime, the dynamics still shows strong autocorrelation, as proved by the appearance of circle, lemniscates or loops in the trajectory. ...

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.

... Using a continuous approximation model, they found that spinning could be maintained even for a vanishing Coriolis force but for parameters non accessible to experimental conditions [25,26]. The same conclusion was drawn by a stability analysis performed with a discrete-time model based on first principles in a specific parameter regime [28,29]. Spin states were observed with walkers released from a harmonic potential but only for limited time [30]. ...

... Walkers undergo parabolic jumps between inelastic bounces. Following Durey et al. [28,29], we consider the bouncing time as instantaneous and the horizontal velocity +1 after the n th bounce constant. The position of the (n+1) th bounce +1 is given by iteration: ...

... We use the discrete iterative path-memory model to assess the stability of the modes [28,29]. ...

Time-varying media can dramatically modify the emission of embedded sources by producing time reversed waves refocusing on the source. Here, we show that such a back action can create an angular momentum by setting the source in a spontaneous spin state. We experimentally implement this coupling using self-propelled bouncing droplets sources coupled to the surface waves they emit on a parametrically excited bath. The spin state dynamics result from a self-organized interplay between the source motion and the time reversed waves. The discrete stability analysis agrees with the experimental observations. In addition, we show that these spin states provide a unique opportunity for an experimental access to parameters enabling comparison and calibration of the various existing models.

... Above a certain fluid acceleration threshold the drop may self-propel and be guided by standing waves [2][3][4][5][6][7][8][9][10][11][12][13] which are the footprint of Faraday waves [14][15][16]. The system has triggered a flurry of thought-provoking experiments which were previously thought to be peculiar to the quantum world [2,3]. ...

... The dynamics of one drop has been investigated in many configurations, like moving through a slit and double slits [26], in cavities [27][28][29][30], with a Coriolis force [31][32][33], in a harmonic potential [13,[34][35][36], exhibiting Friedel-like oscillation [37] and enabling statistical projections [38]. In the last decade, these studies have achieved a converging understanding of both the emergence of wave-like statistics and classical quantization at a macroscopic scale [2,3]. ...

... Two drops dynamics has also been investigated and promenade modes [13,39], as well as quantized orbiting states have been observed and rationalized [1,13,40,41]. As for many drop dynamics, crystalline structures have been obtained in 2D [42,43], and collective dynamics in toroidal channel [44,45] and spatially periodic potentials [46] have been investigated in the linear and non-linear regime [47,48]. ...

A drop bouncing on a vertically-vibrated surface may self-propel forward by Faraday waves and travels along a fluid interface. This system called "walker" forms a non-quantum wave-particle association at a 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 a large number of walkers evolving on a unbounded fluid interface in a presence of a confining potential acting on the particles. We show that even if the individual trajectories are erratic the system presents well-defined ordered internal structure that remains invariant to many parameter variations. We rationalize such non-stationary self-organization thanks to the symmetry of the waves and show that oscillatory pair potentials form a wavy collective state of active matter.

... Experimentally, the system is made of droplets bouncing periodically on an oscillating oil surface. The result is the emergence of a complex standing wave field (14,15) which propels the droplets and also stores information about its past positions (16)(17)(18)(19)(20)(21)(22)(23)(24)(25). In this system, because the droplet slides down the gradient of the local liquid surface, the wavefield acts as a memory that the droplet edits and reads to alter its future dynamics similarly to a Turing machine (26). ...

Information storage, for short "memory", is a key element of autonomous, out-of-equilibrium dynamics, in particular in biological entities. In synthetic active matter, however, the implementation of internal memory in agents is often limited or even absent. As a consequence, most of the investigations in the field of active matter had no choice but to ignore the influence of memory on the dynamics of these systems. We take here the opportunity to explore this question by leveraging one of the very few experimental physical system in which memory can be described in terms of a single and most importantly tunable scalar quantity. Here we consider a particle propelled at a fluid interface by self-generated stationary waves. The amount of souvenirs stored in the wave-memory field can be tuned, allowing for a throughout investigation of the properties of this memory-driven dynamics. We show numerically and experimentally that the accumulation of information in the wave field induces the loss of long-range time correlations. 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 and by evidencing a minimization principle for the wave field.

... Since their discovery in 2005, self-propelled bouncing droplets, called walkers, have drawn much interest because they exhibit an original non-quantum macroscopic wave-particle duality [1][2][3][4][5][6][7] . When placed on a vertically vibrated bath, droplets bounce indefinitely 8 and become selfpropelled, driven by the waves they emit [1][2][3][4] . ...

... Using a continuous approximation model, they found that spinning could be maintained even for vanishing Coriolis force but for parameters non accessible to experimental conditions 24,25 . The same conclusion was drawn by a stability analysis performed with a discrete-time model based on first principles in a specific parameter regime 6,26 . These results seems to contradict experimental observations performed with walkers released from a harmonic potential 27 . ...

... From equations (6) we get : ...

Self-propelled droplets are composed of droplets driven by the waves they emit when bouncing on a vertically vibrated bath. Their dynamics is based on an interplay between the waves and their source. The existence of self-spinning modes is still controversial. Here, we show experimentally that these modes are stable for a class of droplets and emerge spontaneously from noise fluctuations. We perform a discrete stability analysis to confirm experimental observations. In addition, we show that these self-spinning modes provide a unique opportunity for a direct experimental measurement of parameters used in the wave-driven droplet models found in the literature to enable comparison and calibration.

... Durey and Milewski [157] developed a discretetime model for the Faraday wave-droplet interactions on a vibrating bath and estimated the stability of fixed points, with many bifurcations explained through energy considerations. The analysis of bouncing and walking states revealed a Doppler effect, exponential spatial damping, and the travelling capillary wave from a single impact. ...

This article considers additional phenomena that complement the earlier topics addressed by Ibrahim [(Liquid Sloshing Dynamics: Theory and Applications. Cambridge University Press, Cambridge, 2005), (ASME J Fluids Eng 137(9):090801, 2015)]. The first phenomenon is the localized Faraday waves known as oscillons, which were observed in granular materials and liquid layers subjected to parametric excitation. Extreme waves, known as rogue, generated in the Faraday surface ripples, are related to the increase in the horizontal mobility of oscillating solitons (oscillons), and their horizontal motion is random over a limited range of excitation acceleration amplitude. Parametric excitation of water in a Hele–Shaw cell and the associated localized standing surface waves of large amplitude will be discussed. The surface wave pattern exhibited a certain similarity with the three-dimensional axisymmetric oscillon. Faraday waves in superfluid Fermi–Bose mixtures and their wave function will be addressed in terms of position and time as described by the Schrödinger equation with time-dependent parabolic potential. The phenomenon of walking fluid droplets on Faraday waves constitutes the majority portion of this article. Different regimes of droplet motion in terms of droplet physical properties, the fluid bath excitation acceleration amplitude and frequency will be discussed. The droplet trajectory diffraction, when passes through a slit, shares the same random features of electron diffraction. The duality of the droplet-wave field together with the path-memory-driven nonlocality and other related topics will be assessed. This article is complemented with the fascinating phenomenon of the stone and bombs skipping/ricochet over water surface.