# Statistical projection effects in a hydrodynamic pilot-wave system

ABSTRACT :

“Millimetric liquid droplets can walk across the surface of a vibrating fluid bath, self-propelled through a resonant interaction with their own guiding or ‘pilot’ wave fields. These walking droplets, or ‘walkers’, exhibit several features previously thought to be peculiar to the microscopic, quantum realm. In particular, walkers confined to circular corrals manifest a wave-like statistical behaviour reminiscent of that of electrons in quantum corrals. Here we demonstrate that localized topological inhomogeneities in an elliptical corral may lead to resonant projection effects in the walker’s statistics similar to those reported in quantum corrals. Specifically, we show that a submerged circular well may drive the walker to excite specific eigenmodes in the bath that result in drastic changes in the particle’s statistical behaviour. The well tends to attract the walker, leading to a local peak in the walker’s position histogram. By placing the well at one of the foci, a mode with maxima near the foci is preferentially excited, leading to a projection effect in the walker’s position histogram towards the empty focus, an effect strongly reminiscent of the quantum mirage. Finally, we demonstrate that the mean pilot-wave field has the same form as the histogram describing the walker’s statistics.”

Sáenz, P. J., Cristea-Platon, T., & Bush, J. W. (2018). Statistical projection effects in a hydrodynamic pilot-wave system. *Nature Physics*, *14*(3), 315.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2017/12/Saenz-NatPhys-2017-.pdf

# Tunneling with a hydrodynamic pilot-wave model

ABSTRACT : “Eddi et al. [Phys. Rev Lett. 102, 240401 (2009)] presented experimental results demonstrating the unpredictable tunneling of a classical wave-particle association as may arise when a droplet walking across the surface of a vibrating fluid bath approaches a submerged barrier.We here present a theoreticalmodel that captures the influence of bottom topography on this wave-particle association and so enables us to investigate its interaction 2 with barriers. The coupledwave-droplet dynamics results in unpredictable tunneling events.

As reported in the experiments by Eddi et al. and as is the case in quantum tunneling [Gamow, Nature (London) 122, 805 (1928)], the predicted tunneling probability decreases exponentially with increasing barrier width. In the parameter regimes examined, tunnelingbetween two cavities suggests an underlying stationary ergodic process for the droplet’s position.”

Nachbin, A., Milewski, P. A., & Bush, J. W. (2017). Tunneling with a hydrodynamic pilot-wave model. *Physical Review Fluids*, *2*(3), 034801

http://www.fisica.unam.mx/talleres/waves_and_particles/files/Nachbin/NachbinMilewskiBush_PRF17.pdf

# Non-specular reflection of walking droplets

Abstract : Since their discovery by Yves Couder and Emmanuel Fort, droplets walking on a vibrating liquid bath have attracted considerable attention because they unexpectedly exhibit certain features reminiscent of quantum particles. While the behaviour of walking droplets in unbounded geometries has to a large extent been rationalized theoretically, no such rationale exists for their behaviour in the presence of boundaries, as arises in a number of key quantum analogue systems. We here present the results of a combined experimental and theoretical study of the interaction of walking droplets with a submerged planar barrier. Droplets exhibit non-specular reflection, with a small range of reflection angles that is only weakly dependent on the system parameters, including the angle of incidence. The observed behaviour is captured by simulations based on a theoretical model that treats the boundaries as regions of reduced wave speed, and rationalized in terms of momentum considerations.

Pucci, G., Sáenz, P. J., Faria, L. M., & Bush, J. W. (2016). Non-specular reflection of walking droplets. *J Fluid Mech*, *804*, R3.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2016/09/Pucci-JFM-2016.pdf

# Onset of chaos in orbital pilot-wave dynamics

**Abstract :*** We examine the orbital dynamics of droplets self-propelling along the surface of a **vibrating bath. Circular orbital motion may arise when the walking droplet is subjected to one **of three external force fields, the Coriolis force, a simple harmonic force, and a Coulomb **force. Particular attention is given to a theoretical characterization of the onset of chaos that **accompanies the destabilization of such circular orbits.*

Tambasco, L., Harris, D., Oza, A., Rosales, R., & Bush, J. (2015, November). Onset of chaos in orbital pilot-wave dynamics. In *APS Meeting Abstracts*.

# Surface topography measurements of the bouncing droplet experiment

A technical entry by the american team reproducing the synthetic Schlieren free surface measurment already implemented successfully by the Paris Team.

Damiano, A. P., Brun, P. T., Harris, D. M., Galeano-Rios, C. A., & Bush, J. W. (2016). Surface topography measurements of the bouncing droplet experiment. *Experiments in Fluids*, *57*(10), 163.

# Shedding light on pilot-wave phenomena

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,

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

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.”

# Physics Today : letters to the editor

Read More# Faraday pilot wave dynamics : modelling and computation

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.

# Pilot-Wave Hydrodynamics (Annual Review of Fluid Mechanics 2015)

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.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2015/01/Bush-AnnRev2015.pdf

# The wave-induced added mass of walking droplets

Bush, J. W., Oza, A. U., & Moláček, J. (2014). The wave-induced added mass of walking droplets. *Journal of Fluid Mechanics*, *755*, R7.

It has recently been demonstrated that droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic realm. The walker, consisting of a droplet plus its guiding wavefield, is a spatially extended object. We here examine the dependence of the walker mass and momentum on its velocity. Doing so indicates that, when the walker’s time scale of acceleration is long relative to the wave decay time, its dynamics may be described in terms of the mechanics of a particle with a speed-dependent mass and a nonlinear drag force that drives it towards a fixed speed. Drawing an analogy with relativistic mechanics, we define a hydrodynamic boost factor for the walkers. This perspective provides a new rationale for the anomalous orbital radii reported in recent studies

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2014/08/Boost-JFM.pdf

# Generating uniaxial vibration with an electrodynamic shaker and external air bearing

Harris, D. M., & Bush, J. W. (2015). Generating uniaxial vibration with an electrodynamic shaker and external air bearing. *Journal of Sound and Vibration*,*334*, 255-269.

Electrodynamic shakers are widely used in experimental investigations of vibrated fluids and granular materials. However, they are plagued by undesirable internal resonances that can significantly impact the quality of vibration. In this work, we measure the performance of a typical shaker and characterize the influence that a payload has on its performance. We present the details of an improved vibration system based on a concept developed by Goldman (2002) [1] which consists of a typical electrodynamic shaker with an external linear air bearing to more effectively constrain the vibration to a single axis. The principal components and design criteria for such a system are discussed. Measurements characterizing the performance of the system demonstrate considerable improvement over the unmodified test shaker. In particular, the maximum inhomogeneity of the vertical vibration amplitude is reduced from approximately 10 percent to 0.1 percent; moreover, transverse vibrations were effectively eliminated.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2015/01/Harris-Shaker.pdf

# A low‑cost, precise piezoelectric droplet‑on‑demand generator

Harris, D. M., Liu, T., & Bush, J. W. (2015). A low-cost, precise piezoelectric droplet-on-demand generator. *Experiments in Fluids*, *56*(4), 1-7.

We present the design of a piezoelectric droplet-on-demand generator capable of producing droplets of highly repeatable size ranging from 0.5 to 1.4 mm in diameter. The generator is low cost and simple to fabricate. We demonstrate the manner in which droplet diameter can be controlled through variation of the piezoelectric driving waveform parameters, outlet pressure, and nozzle diameter.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2015/04/Harris-DropGenerator.pdf

# Video Lesson – 07/06/2013 – Hydrodynamic Modelling of Pilot-Wave and boucing droplet coupling in a Faraday Problem

“Recent experiments by two groups, Yves Couder (Paris) and John

Bush (MIT) have shown experimentally that droplets will bounce on the

surface of a vertically vibrated bath (instead of coalescing with it),

generating a Faraday-type wavefield at every bounce. From this state, a

pitchfork symmetry breaking bifurcation leads to a “walking” state whereby

the bouncing droplet is “guided” by the self-generated wavefield – the

droplet’s pilot wave. Once this state is achieved a large array of

interesting dynamics ensues with surprising analogies to quantum

mechanical behaviour. We will present a coupled particle-fluid model that

can can be used simulate the dynamics of this problem. This is joint work

with John Bush, Andre Nachbin (IMPA) and Carlos Galeano (IMPA)”

# Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization

Oza, A. U., Harris, D. M., Rosales, R. R., & Bush, J. W. (2014). Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization. *Journal of Fluid Mechanics*, *744*, 404-429.

We present the results of a theoretical investigation of droplets walking on a

rotating vibrating ﬂuid bath. The droplet’s trajectory is described in terms of an

integro-differential equation that incorporates the inﬂuence of its propulsive wave

force. Predictions for the dependence of the orbital radius on the bath’s rotation

rate compare favourably with experimental data and capture the progression from

continuous to quantized orbits as the vibrational acceleration is increased. The orbital

quantization is rationalized by assessing the stability of the orbital solutions, and may

be understood as resulting directly from the dynamic constraint imposed on the drop

by its monochromatic guiding wave. The stability analysis also predicts the existence

of wobbling orbital states reported in recent experiments, and the absence of stable

orbits in the limit of large vibrational forcing

# A Hydrodynamic Pilot Wave Model : RIO 2014

Analysis of currrent state of the art modelling

Reflexions in a 1D Cavity

Computer simulations

(proceedings of RIO 2014 workshop)

http://www.personal.psu.edu/alm24/IMPA/talks/IMPANachbin.pdf

# Droplets walking in a rotating frame: from quantized orbits to multimodal statistics

Harris, D. M., & Bush, J. W. (2014). Droplets walking in a rotating frame: from quantized orbits to multimodal statistics. *Journal of Fluid Mechanics*, *739*, 444-464.

We present the results of an experimental investigation of a droplet walking on the

surface of a vibrating rotating fluid bath. Particular attention is given to demonstrating

that the stable quantized orbits reported by Fort et al. (Proc. Natl Acad. Sci.,

vol. 107, 2010, pp. 17515–17520) arise only for a finite range of vibrational

forcing, above which complex trajectories with multimodal statistics arise. We first

present a detailed characterization of the emergence of orbital quantization, and

then examine the system behaviour at higher driving amplitudes. As the vibrational

forcing is increased progressively, stable circular orbits are succeeded by wobbling

orbits with, in turn, stationary and drifting orbital centres. Subsequently, there is a

transition to wobble-and-leap dynamics, in which wobbling of increasing amplitude

about a stationary centre is punctuated by the orbital centre leaping approximately

half a Faraday wavelength. Finally, in the limit of high vibrational forcing, irregular

trajectories emerge, characterized by a multimodal probability distribution that reflects

the persistent dynamic influence of the unstable orbital states.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2014/01/HB-JFM-2014.pdf

# A trajectory equation for walking droplets: hydrodynamic pilot-wave theory

Oza, A. U., Rosales, R. R., & Bush, J. W. (2013). A trajectory equation for walking droplets: hydrodynamic pilot-wave theory. *Journal of Fluid Mechanics*,*737*, 552-570.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2013/12/ORB-JFM.pdf

Integro-differential equation describing the horizontal motion of a walking droplet

Stability to perturbations

# Drops bouncing on a vibrating bath

Moláček, J., & Bush, J. W. (2013). Drops bouncing on a vibrating bath. Journal of Fluid Mechanics, 727, 582-611.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2013/07/MB1-2013.pdf

Linear and logarithmic spring model of a boucing droplet