The energetics of pilot-wave hydrodynamics

A millimetric droplet may self-propel across the surface of a vertically vibrating liquid bath. The resulting 'walker'. consisting of a droplet dressed in a quasi-monochromatic wave form, exhibits many features previously thought to be exclusive to the quantum realm. While the walker dynamics can be remarkably complex, steady and periodic states arise in which the energy added by the bath vibration necessarily balances that dissipated by viscous effects. The system energetics may then be characterized in terms of the exchange between the bouncing droplet and its guiding or 'pilot' wave. We here characterize this energy exchange theoretically. Specifically, we derive simple formulae characterising the dependence of the droplet's gravitational potential energy and wave energy on the droplet speed. Doing so makes clear the partitioning between the gravitational, wave and kinetic energies of walking droplets in a number of steady, periodic and statistically steady dynamical states. We demonstrate that this partitioning depends exclusively on the ratio of the droplet speed to its speed limit, which yields a beguiling connection to the Lorentz factor in relativistic mechanics.