Extending the droplet-wave statistical correspondence in walking droplet dynamics

Walking droplets-millimetric oil droplets that self-propel across the surface of a vibrating fluid bath-exhibit striking emergent statistics that remain only partially understood. In particular, in a variety of experiments, a robust correspondence has been observed between the droplet's statistical distribution and the time-average of the wave field that guides it. Durey et al. [Chaos 28, 096108 (2018)] rigorously established such a correspondence for single-droplet systems with a single, instantaneous droplet-bath impact during each vibration period, but numerical and experimental evidence suggests that the correspondence should hold far more broadly. Laboratory droplet systems, for instance, often exhibit complex bouncing modes that do not adhere to these hypotheses. We attempt to complete this program in the present work, rigorously extending this statistical correspondence to account for arbitrary droplet-bath impact models, multi-droplet interactions, and non-resonant bouncing. We investigate this correspondence numerically in systems of one and two droplets in 1D geometries, and we highlight how the time-averaged wave field can distinguish between correlated and uncorrelated pairs of droplets.