Pilot-wave dynamics of two identical, in-phase bouncing droplets

Abstract

A droplet bouncing on the surface of a vibrating liquid bath can move horizontally guided by the wave it produces on impacting the bath. The wave itself is modified by the environment, and thus the interactions of the moving droplet with the surroundings are mediated through the wave. This forms an example of a pilot-wave system. Taking the Oza-Rosales-Bush description for walking droplets as a theoretical pilot-wave model, we investigate the dynamics of two interacting identical, in-phase bouncing droplets theoretically and numerically. A remarkably rich range of behaviors is encountered as a function of the two system parameters, the ratio of inertia to drag, κ, and the ratio of wave forcing to drag, β. The droplets typically travel together in a tightly bound pair, although they unbind when the wave forcing is large and inertia is small or inertia is moderately large and wave forcing is moderately small. Bound pairs can exhibit a range of trajectories depending on parameter values, including straight lines, sub-diffusive random walks, and closed loops. The droplets themselves may maintain their relative positions, oscillate towards and away from one another, or interchange positions regularly or chaotically as they travel. We explore these regimes and others and the bifurcations between them through analytic and numerical linear stability analyses and through fully nonlinear numerical simulation.A droplet of liquid can bounce on the surface of a bath of the same liquid indefinitely if the bath is experiencing vertical vibrations in an appropriate range of frequencies. At each bounce, the droplet generates a decaying surface wave. The slope of the wave at the point of the next bounce can impart a horizontal force on the droplet, leading to the droplet walking. Since its discovery 1 in 2005, this experimental system has excited significant research because it can exhibit quantum-like behavior. The droplet is a particle that interacts with its environment through the wave it generates, i.e., it is a pilot-wave system. In this paper we use a previously-published model 2 to explore a two-droplet pilot-wave system. We find a remarkably wide range of behaviors, which we explore in detail.

Lorenz-like systems emerging from an integro-differential trajectory equation of a one-dimensional wave-particle entity

“…This stroboscopic model averages over the droplet's vertical periodic bouncing motion and provides a trajectory equation for its two-dimensional horizontal walking motion by taking into account two key horizontal forces acting on the walker: (i) the horizontal wave force proportional to the gradient of the underlying wave field generated by the walker, and (ii) an effective horizontal drag force composed of aerodynamic drag and momentum loss during impact with the fluid surface. This stroboscopic model rationalizes several hydrodynamic quantum analogs 5,6,16,22,23,[32][33][34][35] and also results in rich dynamical behaviors for walkers [36][37][38][39][40][41][42] .…”

Hydrodynamic quantum field theory: the free particle

“…The walking-droplet system suggests a more general theoretical framework for exploring classical pilot-wave dynamics not accessible in the laboratory [7]. Doing so has led to the discovery of hydrodynamic spin states [28,29], and rich two-particle dynamics [30]. Durey et al [31] examined the stability of the self-propelling state in this general classical pilot-wave framework, showing the propensity for in-line oscillations and emergent statistical behavior with a wavelength corresponding to that of the pilot wave [17].…”

Speed oscillations in classical pilot-wave dynamics

“…It also revealed a parameter regime in which a particle orbiting in an oscillatory potential (with the form of a Bessel function) may achieve a statistically steady state characterized by intermittent switching between unstable circular orbits [17]. The same framework was adopted by Valani & Slim [18], who discovered an abundance of exotic dynamical states for two interacting walkers.…”