A hydrodynamic analog of the quantum potential*

A droplet may walk on a vibrating fluid bath through a resonant interaction with its own wave field. This walking droplet system has become the subject of research in the nascent field of hydrodynamic quantum analogs. We here consider the motion of droplets walking in closed systems, and demonstrate the relation between the histogram of the particle and its mean pilot-wave field. Furthermore, we demonstrate that as the Faraday threshold is approached, the instantaneous wave field converges to its mean. The resulting mean pilot-wave potential thus plays the role of the quantum potential in Bohmian mechanics. Our study highlights the differences between Bohmian mechanics and de Broglie's relatively rich double-solution theory of quantum dynamics.