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

Abstract

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.

Dynamics and statistics of wave-particle interactions in a confined geometry

“…For β = π/3, this happens at µ 0.917. Similarly in (i), the wobbling amplitude is shown to increase with memory until the wobbling orbit collides with the most adjacent unstable orbit [9]. At this stage, a chaotic attractor is revealed as no other stable regular attractor is left (Fig.3c, with µ = 0.9369).…”

Simulations of pilot-wave dynamics in a simple harmonic potential

“…In future work, one may apply the methodology of Durey and Milewski [26] followed here to the chaotic trajectories of walkers under the influence of a Coriolis force [12,23,39] with a view to seeking a similar double quantization in that system. A broader exploration of pilot-wave systems with different external forces and geometries should yield a better understanding of the emergent statistics of chaotic pilot-wave dynamics.…”

“…While claims of single-particle diffraction and interference [6] have been contested [7], tunneling [8,9], orbital quantization in both a rotating frame [10][11][12] and harmonic potential [13,14], and wavelike statistics in confined geometries [15][16][17][18] are all robust quantumlike phenomena. This hydrodynamic pilot-wave system and its relation to realist quantum theories, specifically, de Broglie's double-solution pilot-wave theory [19] and its modern extensions [20], were recently reviewed by Bush [21,22].…”

“…Oza et al [11] and Harris and Bush [12] revisited this system theoretically and experimentally, respectively. Linear stability analysis of the orbital pilot-wave solutions provided rationale for the emergence of orbital quantization, as was manifest in bands of accessible orbital radii [11].…”

The new wave of pilot-wave theory

“…A few years later, my colleagues and I examined the high-memory regime of the rotating system both experimentally 11 and theoretically. 12 At high memory, virtually all orbital states become unstable, typically via a period-doubling transition to chaos.…”