Walking droplets interacting with planar boundaries

Journal of Fluid Mechanics, 2020

A millimetric droplet of silicone oil may bounce and self-propel on the free surface of a vertically vibrating fluid bath due to the droplet's interaction with its accompanying Faraday wave field. This hydrodynamic pilot-wave system exhibits many dynamics that were previously thought to be peculiar to the quantum realm. When the droplet is confined to a circular cavity, referred to as a 'corral', a range of dynamics may occur depending on the details of the geometry and the decay time of the subcritical Faraday waves. We herein present a theoretical investigation into the behaviour of subcritical Faraday waves in this geometry and explore the accompanying pilot-wave dynamics. By computing the Dirichlet-to-Neumann map for the velocity potential in the corral geometry, we can evolve the quasi-potential flow between successive droplet impacts, which, when coupled with a simplified model for the droplet's vertical motion, allows us to derive and implement a highly efficient discrete-time iterative map for the pilot-wave system. We study the onset of the Faraday instability, the emergence and quantisation of circular orbits and simulate the exotic dynamics that arises in smaller corrals.

Physical review fluids, 2017

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 theoretical model that captures the influence of bottom topography on this wave-particle association and so enables us to investigate its interaction with barriers. The coupled wave-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, tunneling between two cavities suggests an underlying stationary ergodic process for the droplet's position.

Comptes Rendus. Mécanique

We revisit de Broglie's double-solution pilot-wave theory in light of insights gained from the hydrodynamic pilot-wave system discovered by Couder and Fort [1]. de Broglie proposed that quantum particles are characterized by an internal oscillation at the Compton frequency, at which rest mass energy is exchanged with field energy. He further proposed that the resulting pilot-wave field satisfies the Klein-Gordon equation. While he developed a guidance equation for the particle, he did not specify how the particle generates the wave. Informed by the hydrodynamic pilot-wave system, we explore a variant of de Broglie's mechanics in which the form of the Compton-scale dynamic interaction between particle and pilot wave is specified. The particle is modeled as a localized periodic disturbance of the Klein-Gordon field at twice the Compton frequency. We simulate the evolution of the particle position by assuming that the particle is propelled by the local gradient of its pilot wave field. Resonance is achieved between the particle and its pilot wave, leading to self-excited motion of the particle. The particle locks into quasi-steady motion characterized by a mean momentump = ħk, where k is the wavenumber of the surrounding matter waves. Speed modulations along the particle path arise with the de Broglie wavelength and frequency ck. The emergent dynamics is strongly reminiscent of that arising in the hydrodynamic pilot-wave system, on the basis of which we anticipate the emergence of quantum statistics in various settings. Our results suggest the potential value of a new hydrodynamically-inspired pilot-wave theory for the motion of quantum particles.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2020

Recent experiments show that quasi-one-dimensional lattices of self-propelled droplets exhibit collective instabilities in the form of out-of-phase oscillations and solitary-like waves. This hydrodynamic lattice is driven by the external forcing of a vertically vibrating fluid bath, which invokes a field of subcritical Faraday waves on the bath surface, mediating the spatio-temporal droplet coupling. By modelling the droplet lattice as a memory-endowed system with spatially non-local coupling, we herein rationalize the form and onset of instability in this new class of dynamical oscillator. We identify the memory-driven instability of the lattice as a function of the number of droplets, and determine equispaced lattice configurations precluded by geometrical constraints. Each memory-driven instability is then classified as either a super- or subcritical Hopf bifurcation via a systematic weakly nonlinear analysis, rationalizing experimental observations. We further discover a previou...

Physical Review Letters

Superradiance occurs when a collection of atoms exhibits cooperative, spontaneous emission of photons at a rate that exceeds that of its component parts. Here, we reveal a similar phenomenon in a hydrodynamic system consisting of a pair of vibrationally-excited cavities, coupled through their common wavefield, that spontaneously emit droplets via interfacial fracture. We show that the droplet emission rate of two coupled cavities is higher than the emission rate of two isolated cavities. Moreover, the amplified emission rate varies sinusoidally with distance between the cavities, as is characteristic of superradiance. We thus present a hydrodynamic phenomenon that captures several essential features of superradiance in optical systems.

Nature, 2021

single-spin measurements revealed an optimal signal-to-driving ratio for S(t) at a vibrational acceleration γ c ≈ 0.85γ F (Fig. 1c). For suboptimal driving γ < γ c , the mean spin decreases with the walker's characteristic speed as γ decreases 36. For γ > γ c , the circular orbits deform into precessing trefoils (Fig. 1c) that exhibit a larger spin variance, making them more susceptible to spin flips when perturbed. Armed with this understanding of the single-spin states, we investigated the collective spin dynamics in one-dimensional (1D) and two-dimensional (2D) HSLs. The wave-mediated spin-spin coupling in HSLs is reminiscent of spatially oscillating interactions 9,38 in Ruderman-Kittel-Kasuya-Yosida (RKKY)-type spin models. Our experimental setup allowed us to tune the magnitude of the spin-spin coupling by varying the driving acceleration γ and the depth H of the fluid bath between adjacent wells (Fig. 1b; Methods section 'Experiments'). When the pair coupling is sufficiently strong, nearest-neighbour interactions may cause spin flips. To determine whether such flips can facilitate coherent collective dynamics across the lattice, we measured the normalized effective 'magnetization' M(t) = ∑ i S i (t) and spin-spin correlation χ(t) = ∑ i~j S i (t)S j (t), where ∑ i~j denotes a sum over adjacent pairs. To achieve statistical significance, experiments were run for several hours: in 1 h a droplet performs approximately 10 5 bounces and approximately 1,800 orbits (Methods section 'Statistics'). Positive values of χ signal parallel alignment of neighbouring spins ('ferromagnetic' order) whereas negative values of χ indicate antiparallel alignment ('antiferromagnetic' order). Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Abstract We present a new simulation method to calculate the free energy and the chemical potential of a hard particle system and investigate the isotropic-nematic phase transition. The method relies on the introduction of a parameter dependent potential to smoothly transform between the hard particle system and the corresponding ideal gas. We applied the method to study the phase transition behavior of a square monodispersed platelet system. The equilibrium state was found with an isobaric Monte Carlo (MC) technique.

Physics of Fluids, 2013

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2018

Millimetric droplets may be levitated on the surface of a vibrating fluid bath. Eddi et al. [Europhys. Lett. 82, 44001 (2008)] demonstrated that when a pair of levitating drops of unequal size are placed nearby, they interact through their common wavefield in such a way as to self-propel through a ratcheting mechanism. We present the results of an integrated experimental and theoretical investigation of such ratcheting pairs. Particular attention is given to characterizing the dependence of the ratcheting behavior on the droplet sizes and vibrational acceleration. Our experiments demonstrate that the quantized inter-drop distances of a ratcheting pair depend on the vibrational acceleration, and that as this acceleration is increased progressively, the direction of the ratcheting motion may reverse up to four times. Our simulations highlight the critical role of both the vertical bouncing dynamics of the individual drops and the traveling wave fronts generated during impact on the ra...

Physical Review E, 2013

Bouncing droplets can self-propel laterally along the surface of a vibrated fluid bath by virtue of a resonant interaction with their own wave field. The resulting walking droplets exhibit features reminiscent of microscopic quantum particles. Here we present the results of an experimental investigation of droplets walking in a circular corral. We demonstrate that a coherent wavelike statistical behavior emerges from the complex underlying dynamics and that the probability distribution is prescribed by the Faraday wave mode of the corral. The statistical behavior of the walking droplets is demonstrated to be analogous to that of electrons in quantum corrals.