Absence of diffusion in pilot-wave hydrodynamics: A classical analog of Anderson localization

A self-propelled particle in heterogeneous media may undergo diffusive motion when its energy is comparable to the average potential barrier of the random background. The evolution of an ensemble of such particles may thus be described by the diffusion equation, leading to a uniform spatial distribution in the long-time limit. In this talk, we will introduce a hydrodynamic pilot-wave system in which particles instantaneously exhibit diffusive motion, yet their position histogram becomes localized. The constituents of this hydrodynamic pilot-wave system are millimetric liquid droplets that may walk across the surface of a vibrating fluid bath, self-propelled through a resonant interaction with their own wave fields. By virtue of the coupling with their wave fields, these walking droplets, or 'walkers', extend the range of classical mechanics to include certain features previously thought to be exclusive to the microscopic, quantum realm. Through experiments and mathematical modeling, we investigate the motion of walkers over submerged random topographies. For sufficiently shallow liquid layers, the walker trajectory becomes chaotic due to scattering from random features at the bottom of the bath. Nevertheless, consideration of an ensemble of drop trajectories reveals that our hydrodynamic pilot-wave system displays localized statistics in the particle position histogram, an effect strongly reminiscent to the so-called Anderson localization. Particular attention is given to characterizing the influence of the submerged topography on the emergent particle dynamics and long-time probability distributions. The localized statistics are rationalized in terms of a wave-mediated scattering mechanism.