Quantumlike statistics of deterministic wave-particle interactions in a circular cavity

Abstract

A deterministic low-dimensional iterated map is proposed here to describe the interaction between a bouncing droplet and Faraday waves 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 Schrödinger equation for a quantum particle in a similar potential well.

Two-frequency forcing of droplet rebounds on a liquid bath

“…Oscillations of the horizontal velocity at the scale of the Faraday wavelength have already been observed for walkers in several configurations [13,14,36]. In particular, we showed in a recent work that for walkers confined in cavities, the Faraday wavelength is the elementary step of a 2D random walk [37,38] which statistics is shaped by the wave in accordance with Schrödinger equation. In the present work, the walker also explores the bath surface with steps of approximate size λ 4−5 (Fig.…”

“…11d and 11h). The average angular velocity of the droplet is around 0.08 rad/s, so any diffusion-like behavior could only be observed at a timescale larger than 2π/0.08 79 s [38]. Unfortunately, our recording time was limited to 55 s so the diffusion behavior could not be evidenced.…”

“…The distance D increases almost linearly with time T so the motion is ballistic (of average speed 5.7 mm/s), at least on timescales of a few seconds [38]. Indeed, the walker does not change direction at every step (e.g.…”

“…In previous works on walkers, the chaos in the horizontal trajectory always resulted from the confinement of the particle and associated wave. The confinement was ensured by a central force [14,39], by a Coriolis force [ 12,40], or by physical boundaries [13,38]. In each case, the vertical trajectory could be assumed periodic.…”

Simulations of pilot-wave dynamics in a simple harmonic potential

“…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].…”