On this video you will see how a walking droplet in a small 1D cavity moves “randomly” if the memory of the system is high enough (ie if the forcing is strong enough, but still below the Faraday Thresold)

Gilet, T. (2016). Quantumlike statistics of deterministic wave-particle interactions in a circular cavity. Physical Review E, 93(4), 042202.

A deterministic low-dimensional iterated map is proposed here to describe the interaction between a bouncing droplet and Faradaywaves 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 /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 Schr¨odinger equation for a quantum particle in a similar potential well.

These pictures illustrate path memory : in the wake of the drop, there is a superposition of a circular wave due to latest impact and of “line waves” created by the many previous bounces

Russian Physicians from Moscow Institute of Physics and Technology have chosen a dotwave.org picture to illustrate an article published on phys.org concerning their latest paper,

Des physiciens russes de l’institut de physique et de technologie de Moscou ont choisi une de mes photos pour illustrer un résumé d’un de leur papier sur les ondes de Faraday publié sur phys.org

Domino, L., Tarpin, M., Patinet, S., & Eddi, A. (2016). Faraday wave lattice as an elastic metamaterial. arXiv preprint arXiv:1601.08024.

(Also on PhysRev E.)

Metamaterials enable the emergence of novel physical properties due to the existence of an underlying sub-wavelength structure. Here, we use the Faraday instability to shape the uid-air interface with a regular pattern. This pattern undergoes an oscillating secondary instability and exhibits spontaneous vibrations that are analogous to transverse elastic waves. By locally forcing these waves, we fully characterize their dispersion relation and show that a Faraday pattern presents an
effective shear elasticity. We propose a physical mechanism combining surface tension with the Faraday structured interface that quantitatively predicts the elastic wave phase speed, revealing that the liquid interface behaves as an elastic metamaterial.

Dubertrand, R., Hubert, M., Schlagheck, P., Vandewalle, N., Bastin, T., & Martin, J. (2016). Scattering theory of walking droplets in the presence of obstacles. arXiv preprint arXiv:1605.02370.

We aim to describe a droplet bouncing on a vibrating bath. Due to Faraday instability a surface wave is created at each bounce and serves as a pilot wave of the droplet. This leads to so called walking droplets or walkers. Since the seminal experiment by Couder et al [Phys. Rev. Lett. 97, 154101 (2006)] there have been many attempts to accurately reproduce the experimental results. Here we present a simple and highly versatile model inspired from quantum mechanics. We propose to describe the trajectories of a walker using a Green function approach. The Green function is related to Helmholtz equation with Neumann boundary conditions on the
obstacle(s) and outgoing conditions at infinity. For a single slit geometry our model is exactly solvable and reproduces some general features observed experimentally. It
stands for a promising.

Filatov, S. V., Parfenyev, V. M., Vergeles, S. S., Brazhnikov, M. Y., Levchenko, A. A., & Lebedev, V. V. (2016). Nonlinear Generation of Vorticity by Surface Waves. Physical review letters, 116(5), 054501.

We demonstrate that waves excited on a fluid surface produce local surface rotation owing to hydrodynamic nonlinearity. We examine theoretically the effect and obtain an explicit formula for the vertical vorticity in terms of the surface elevation. Our theoretical predictions are confirmed by measurements of surface motion in a cell with water where surface waves are excited by vertical and harmonic shaking the cell. The experimental data are in good agreement with the theoretical predictions. We discuss physical consequences of the effect.

Labousse, M., Oza, A. U., Perrard, S., & Bush, J. W. (2016). Pilot-wave dynamics in a harmonic potential: Quantization and stability of circular orbits.Physical Review E, 93(3), 033122.

“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.”

Milewski, P. A., Galeano-Rios, C. A., Nachbin, A., & Bush, J. W. (2015). Faraday pilot-wave dynamics: modelling and computation. Journal of Fluid Mechanics, 778, 361-388.

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 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.

Gilet, T. (2014). Dynamics and statistics of wave-particle interactions in a confined geometry. Physical Review E, 90(5), 052917.

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 2D 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.

Brandenbourger, M., Vandewalle, N., & Dorbolo, S. (2016). Displacement of an Electrically Charged Drop on a Vibrating Bath. Physical review letters, 116(4), 044501.

In this work, the manipulation of an electrically charged droplet bouncing on a vertically vibrated, bath is investigated. When a horizontal, uniform and static electric eld is applied to it, a motion is induced. The droplet is accelerated when the droplet is small. On the other hand, large droplets appear to move with a constant speed that depends linearly on the applied electrical eld. In the latter regime, high speed imaging of one bounce reveals that the droplet experiences an acceleration due to the electrical force during the ight and decelerates to zero when interacting with the surface of the bath. Thus, the droplet moves with a constant average speed on a large time scale. We propose a criterion based on the force necessary to move a charged droplet at the surface of the
bath to discriminate between constant speed and accelerated droplet regimes.

A walking droplet is placed in a square box, at the onset of Faraday thresold.

The trajectory of the droplet is mapped.
In the long time limit, does a self-interference pattern appear ? what’s its shape ? How does it relate to the square cavity surface wave eigen-modes ?

In short, we try to reproduce the experiment of Bush et al, but in a square box.

First result :

A walking droplet in a square cavity shows random motion, but with time, its trajectory is building a statistic reminiscent of the resonant mode of the cavity.

This can be seen by the naked eye in this movie excerpt :

This is then confirmed with optical tracking measurment of the trajectory :

Trajectory of the walking droplet

The position distribution (~probability density) is then computed :

I was lucky enough to attend this mini-colloque !!

“Si la dualité onde-corpuscule est une des bases de l’interprétation de la mécanique quantique, elle peut aussi se manifester à l’échelle macroscopique. Durant ce mini-colloque seront présentées les propriétés observées dans des systèmes macroscopiques, ainsi que quelques-unes de leurs pendants aux échelles microscopiques. Un des objectifs est d’identifier les analogies et les différences entre ces deux types de systèmes.”

Molteni, D., Vitanza, E., & Battaglia, O. R. (2016). Smoothed Particles Hydrodynamics numerical simulations of droplets walking on viscous vibrating fluid. arXiv preprint arXiv:1601.05017.

Abstract :

“We study the phenomenon of the “walking droplet”, by means of numerical fluid dynamics simulations using a standard version of the Smoothed Particle Hydrodynamics method. The phenomenon occurs when a millimetric drop is released on the surface of an oil of the same composition contained in a container subjected to vertical oscillations of frequency and amplitude close to the Faraday instability threshold. At appropriate values of the parameters of the system under study, the liquid drop jumps permanently on the surface of the vibrating fluid forming a localized wave-particle system, reminding the behavior of a wave particle quantum system as suggested by de Broglie. In the simulations, the drop and the wave travel at nearly constant speed, as observed in experiments. In our study we made relevant simplifying assumptions, however we observe that the wave-drop coupling is easily obtained. This fact suggests that the phenomenon may occur in many contexts and opens the possibility to study the phenomenon in an extremely wide range of physical configurations.”

Bacot, V., Labousse, M., Eddi, A., Fink, M., & Fort, E. (2015). Revisiting time reversal and holography with spacetime transformations. arXiv preprint arXiv:1510.01277.

Wave control is usually performed by spatially engineering the properties of a medium. Because time and space play similar roles in wave propagation, manipulating time boundaries provides a complementary approach. Here, we experimentally demonstrate the relevance of this concept by introducing instantaneous time mirrors. We show with water waves that a sudden change of the effective gravity generates time-reversed waves that refocus at the source. We generalize this concept for all kinds of waves introducing a universal framework which explains the effect of any time disruption on wave propagation. We show that sudden changes of the medium properties generate instant wave sources that emerge instantaneously from the entire space at the time disruption. The time-reversed waves originate from these “Cauchy sources” which are the counterpart of Huygens virtual sources on a time boundary. It allows us to revisit the holographic method and introduce a new approach for wave control.

Filoux, B., Hubert, M., Schlagheck, P., & Vandewalle, N. (2015). Waveguides for walking droplets. arXiv preprint arXiv:1507.08228.

When gently placing a droplet onto a vertically vibrated bath, a drop can bounce permanently. Upon increasing the forcing acceleration, the droplet is propelled by the wave it generates and becomes a walker with a well dened speed. We investigate the connement of a walker in different rectangular cavities, used as waveguides for the Faraday waves emitted by successive droplet bounces. By studying the walker velocities, we discover that 1d connement is optimal for narrow channels. We also propose an analogy with waveguide models based on the observation of the Faraday instability within the channels.

Bush, J. W. (2015). Pilot-wave hydrodynamics. Annual Review of Fluid Mechanics, 47, 269-292.

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 quantumlike 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.