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arXiv:1401.4356

Why bouncing droplets are a pretty good model of quantum mechanics

Robert M. Brady, Ross Anderson

Published Jan 16, 2014• 12 citations
arXivHTMLPDF

Abstract

In 2005, Couder, Protiere, Fort and Badouad showed that oil droplets bouncing on a vibrating tray of oil can display nonlocal interactions reminiscent of the particle-wave associations in quantum mechanics; in particular they can move, attract, repel and orbit each other. Subsequent experimental work by Couder, Fort, Protiere, Eddi, Sultan, Moukhtar, Rossi, Molacek, Bush and Sbitnev has established that bouncing drops exhibit single-slit and double-slit diffraction, tunnelling, quantised energy levels, Anderson localisation and the creation/annihilation of droplet/bubble pairs. In this paper we explain why. We show first that the surface waves guiding the droplets are Lorentz covariant with the characteristic speed c of the surface waves; second, that pairs of bouncing droplets experience an inverse-square force of attraction or repulsion according to their relative phase, and an analogue of the magnetic force; third, that bouncing droplets are governed by an analogue of Schroedinger's equation where Planck's constant is replaced by an appropriate constant of the motion; and fourth, that orbiting droplet pairs exhibit spin-half symmetry and align antisymmetrically as in the Pauli exclusion principle. Our analysis explains the similarities between bouncing-droplet experiments and the behaviour of quantum-mechanical particles. It also enables us to highlight some differences, and to predict some surprising phenomena that can be tested in feasible experiments.

DOI
10.48550/arxiv.1401.4356

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A millimetric droplet can be induced to bounce on the surface of a fluid bath by vibrating the bath near the droplet's resonant frequency (Figure 1(a) ). [superscript 1–3] The localized field of Faraday waves excited by the bouncing droplet can cause it to propel itself laterally across the surface, moving in resonance with its guiding wave field (Figure 1(b) ). [superscript 4,5] These walking droplets, or “walkers,” generally move in a straight line at constant speed; however, they can be diverted through interaction with boundaries or external forces. This hydrodynamic system represents a macroscopic realization of the pilot-wave theory of quantum dynamics proposed by Louis de Broglie, according to which microscopic particles are propelled through a resonant interaction with a wave field generated by the particle's internal vibration. [superscript 6] Coincidentally, it exhibits many behaviors once thought to be exclusive to the microscopic quantum realm, including single-particle diffraction, [superscript 7] tunneling, [superscript 8] quantized orbits, [superscript 9] and orbital-level splitting. [superscript 10]

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In 2005, Couder, Protiere, Fort and Badouad showed that oil droplets bouncing on a vibrating tray of oil can display nonlocal interactions reminiscent of the particle-wave associations in quantum mechanics; in particular they can move, attract, repel and orbit each other. Subsequent experimental work by Couder, Fort, Protiere, Eddi, Sultan, Moukhtar, Rossi, Molacek, Bush and Sbitnev has established that bouncing drops exhibit single-slit and double-slit diffraction, tunnelling, quantised energy levels, Anderson localisation and the creation/annihilation of droplet/bubble pairs. In this paper we explain why. We show first that the surface waves guiding the droplets are Lorentz covariant with the characteristic speed c of the surface waves; second, that pairs of bouncing droplets experience an inverse-square force of attraction or repulsion according to their relative phase, and an analogue of the magnetic force; third, that bouncing droplets are governed by an analogue of Schroedinger's equation where Planck's constant is replaced by an appropriate constant of the motion; and fourth, that orbiting droplet pairs exhibit spin-half symmetry and align antisymmetrically as in the Pauli exclusion principle. Our analysis explains the similarities between bouncing-droplet experiments and the behaviour of quantum-mechanical particles. It also enables us to highlight some differences, and to predict some surprising phenomena that can be tested in feasible experiments.
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.
We explore the hydrodynamic analogues of quantum wave-particle duality in the context of a bouncing droplet system which we model in such a way as to promote comparisons to the de Broglie-Bohm interpretation of quantum mechanics. Through numerical means we obtain single-slit diffraction and double-slit interference patterns that strongly resemble those reported in experiment and that reflect a striking resemblance to quantum diffraction and interference on a phenomenological level. We, however, identify evident differences from quantum mechanics which arise from the governing equations at the fundamental level.
In a pioneering series of experiments, Yves Couder, Emmanuel Fort and coworkers demonstrated that droplets bouncing on the surface of a vertically vibrating fluid bath exhibit phenomena reminiscent of those observed in the microscopic quantum realm. Inspired by this discovery, we here conduct a theoretical and numerical investigation into the structure and dynamics of one-dimensional chains of bouncing droplets. We demonstrate that such chains undergo an oscillatory instability as the system’s wave-induced memory is increased progressively. The predicted oscillation frequency compares well with previously reported experimental data. We then investigate the resonant oscillations excited in the chain when the drop at one end is subjected to periodic forcing in the horizontal direction. At relatively high memory, the drops may oscillate with an amplitude larger than that prescribed, suggesting that the drops effectively extract energy from the collective wave field. We also find that dynamic stabilization of new bouncing states can be achieved by forcing the chain at high frequency. Generally, our work provides insight into the collective behavior of particles interacting through long-range and temporally nonlocal forces.
Couder & Fort (Phys. Rev. Lett., vol. 97, 2006, 154101) demonstrated that when a droplet walking on the surface of a vibrating bath passes through a single or a double slit, it is deflected due to the distortion of its guiding wave field. Moreover, they suggested the build-up of statistical diffraction and interference patterns similar to those arising for quantum particles. Recently, these results have been revisited (Andersen et al., Phys. Rev. E, vol. 92 (1), 2015, 013006; Batelaan et al., J. Phys.: Conf. Ser., vol. 701 (1), 2016, 012007) and contested (Andersen et al. 2015; Bohr, Andersen & Lautrup, Recent Advances in Fluid Dynamics with Environmental Applications, 2016, Springer, pp. 335–349). We revisit these experiments with a refined experimental set-up that allows us to systematically characterize the dependence of the dynamical and statistical behaviour on the system parameters. The system behaviour is shown to depend strongly on the amplitude of the vibrational forcing: as this forcing increases, a transition from repeatable to unpredictable trajectories arises. In all cases considered, the system behaviour is dominated by a wall effect, specifically the tendency for a drop to walk along a path that makes a fixed angle relative to the plane of the slits. While the three dominant central peaks apparent in the histograms of the deflection angle reported by Couder & Fort (2006) are evident in some of the parameter regimes considered in our study, the Fraunhofer-like dependence of the number of peaks on the slit width is not recovered. In the double-slit geometry, the droplet is influenced by both slits by virtue of the spatial extent of its guiding wave field. The experimental behaviour is well captured by a recently developed theoretical model that allows for a robust treatment of walking droplets interacting with boundaries. Our study underscores the importance of experimental precision in obtaining reproducible data.
Just over a decade ago Couder and Fort [Phys. Rev. Lett. 97, 154101 (2006)PRLTAO0031-900710.1103/PhysRevLett.97.154101] published a provocative paper suggesting that a classical system might be able to simulate the truly fundamental quantum mechanical single- and double-slit experiment. The system they investigated was that of an oil droplet walking on a vibrated oil surface. Their results have since been challenged by Andersen et al. [Phys. Rev. E 92, 013006 (2015)PLEEE81539-375510.1103/PhysRevE.92.013006] by pointing to insufficient statistical support and a lack of experimental control over critical parameters. Here we show that the randomness in the original experiment is an artifact of lack of control. We present experimental data from an extensive scan of the parameter space of the system including the use of different size slits and tight control of critical parameters. For the single-slit we find very diverse samples of interference-like patterns but all causal by nature. This also holds for the double-slit. However, an extra interference effect appears here. The origin of this is investigated by blocking either the inlet or the outlet of one slit. Hereby we show that the extra interference is solely due to back-scatter of the associated wave field from the outlet of the slit not passed by the droplet. Recently Pucci et al. [J. Fluid Mech. 835, 1136 (2018)JFLSA70022-112010.1017/jfm.2017.790] using a much broader slit also showed that the classical system is basically causal. They, too, observed the extra interference effect for the double-slit. However, the reason behind was not determined. Moreover they claimed the existence of a chaotic regime just below the cri- tical acceleration for spontaneous generation of Faraday surface waves. Our measurements do not support the validity of this claim. However, the drop dynamics turns out to have an interesting multifaceted interaction with the slit structure.
A millimetric droplet can be induced to bounce on the surface of a fluid bath by vibrating the bath near the droplet's resonant frequency (Figure 1(a) ). [superscript 1–3] The localized field of Faraday waves excited by the bouncing droplet can cause it to propel itself laterally across the surface, moving in resonance with its guiding wave field (Figure 1(b) ). [superscript 4,5] These walking droplets, or “walkers,” generally move in a straight line at constant speed; however, they can be diverted through interaction with boundaries or external forces. This hydrodynamic system represents a macroscopic realization of the pilot-wave theory of quantum dynamics proposed by Louis de Broglie, according to which microscopic particles are propelled through a resonant interaction with a wave field generated by the particle's internal vibration. [superscript 6] Coincidentally, it exhibits many behaviors once thought to be exclusive to the microscopic quantum realm, including single-particle diffraction, [superscript 7] tunneling, [superscript 8] quantized orbits, [superscript 9] and orbital-level splitting. [superscript 10]