Damped-driven system of bouncing droplets leading to deterministically diffusive behavior

Abstract

Damped-driven systems are ubiquitous in science, however the damping and driving mechanisms are often quite convoluted. This manuscript presents an experimental and theoretical investigation of a fluidic droplet on a vertically vibrating fluid bath as a damped-driven system. We study a fluidic droplet in an annular cavity with the fluid bath forced above the Faraday wave threshold. We model the droplet as a kinematic point particle in air and as inelastic collisions during impact with the bath. In both experiments and the model the droplet is observed to chaotically change velocity with a Gaussian distribution leading to diffusion-like behavior. In addition, the forcing above the Faraday wave threshold on the fluid bath simplifies the wave dynamics to that of a standing wave, which allows us to explicitly segregate the damping and driving mechanisms. The energy gain comes from the kinematics of the droplet between impacts, and the energy loss comes from the hydrodynamic damping at impact. We show that this energy gain-loss formulation exhibits dynamical behavior canonical to a wide range of damped-driven systems. The bifurcations and route to chaos present in the droplet system reveals analogies with other well-studied systems from optics, detonation, and electronics. This also hints at analogies with many more systems that have yet to be analyzed in detail, and paves a framework with which other systems can be analyzed. Finally, the statistical distributions from experiments and theory are analyzed. Incredibly, this simple deterministic interaction of damping and driving of the droplet leads to more complex Brownian-like and Levy-like behavior.