Brun, P. T., Harris, D. M., Prost, V., Quintela, J., & Bush, J. W. (2016). Shedding light on pilot-wave phenomena. Physical Review Fluids, 1(5), 050510.

ABSTRACT

This paper is associated with a video winner of a 2015 APS/DFD Gallery of Fluid Motion Award. The original video is available from the Gallery of Fluid Motion,

Rahman, A., & Blackmore, D. (2015). Neimark–Sacker bifurcation and evidence of chaos in a discrete dynamical model of walkers. arXiv preprint arXiv:1507.08057.

ABSTRACT

Bouncing droplets on a vibrating fluid bath can exhibit wave-particle behavior, such as being propelled by interacting with its own wave field. These droplets seem to walk across the bath, and thus are dubbed walkers. Experiments have shown that walkers can exhibit exotic dynamical behavior indicative of chaos. While the integro-differential models developed for these systems agree well with the experiments, they are difficult to analyze mathematically. In recent years, simpler discrete dynamical models have been derived and studied numerically. The numerical simulations of these models show evidence of exotic dynamics such as period doubling bifurcations, Neimark–Sacker (N–S) bifurcations, and even chaos. For example, in [Gilet, PRE 2014], based on simulations Gilet conjectured the existence of a supercritical N-S bifurcation as the damping factor in his one-dimensional path model. We prove Gilet’s conjecture and more; in fact, both supercritical and subcritical (N-S) bifurcations are produced by separately varying the damping factor and wave-particle coupling for all eigenmode shapes. Then we compare our theoretical results with some previous and new numerical simulations, and find complete qualitative agreement. Furthermore, evidence of chaos is shown by numerically studying a global bifurcation.

Perrard, S., Fort, E., & Couder, Y. (2016). Wave-Based Turing Machine: Time Reversal and Information Erasing. Physical Review Letters, 117(9), 094502.

ABSTRACT

The investigation of dynamical systems has revealed a deep-rooted difference between waves and objects regarding temporal reversibility and particlelike objects. In nondissipative chaos, the dynamic of waves always remains time reversible, unlike that of particles. Here, we explore the dynamics of a wave-particle entity. It consists in a drop bouncing on a vibrated liquid bath, self-propelled and piloted by the surface waves it generates. This walker, in which there is an information exchange between the particle and the wave, can be analyzed in terms of a Turing machine with waves as the information repository. The experiments reveal that in this system, the drop can read information backwards while erasing it. The drop can thus backtrack on its previous trajectory. A transient temporal reversibility, restricted to the drop motion, is obtained in spite of the system being both dissipative and chaotic.

Parker, J. (2015). Transition Orbits of Walking Droplets (Doctoral dissertation, California Polytechnic State University, San Luis Obispo).

“It was recently discovered that millimeter-sized droplets bouncing on the surface of an oscillating bath of the same fluid can couple with the surface waves it produces and begin walking across the fluid bath. These walkers have been shown to behave similarly to quantum particles; a few examples include single-particle diffraction, tunneling, and quantized orbits. Such behavior occurs because the drop and surface waves depend on each other to exist, making this the first and only known macroscopic pilot-wave system. In this paper, the quantized orbits between two identical drops are explored. By sending a perturbation to a pair of orbiting walkers, the orbit can be disrupted and transition to a new orbit. The numerical results of such transitions are analyzed and discussed.”

TWO interesting things in this small report :

1 – Influence of apparatus temperature on the faraday thresold

2 – sending a plane wave towards a 2-droplet orbiting system can cause a shift in the orbit quantization

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.”