From Walking to Tunneling: An Investigation of Generalized Pilot-Wave Dynamics

We investigate the ability of millimetric walking droplets to tunnel between spatially-structured cavities. By synthesizing experimental and theoretical analysis, we provide a comprehensive framework for droplet tunneling mechanics in three spatial dimensions. We define a generalized Dirichlet-to-Neumann operator that enables explicit characterization of droplet and wave-field dynamics under highly intricate variable-topography systems. This formalism enables a reduced, three-dimensional description of the pilot-wave field, facilitating high-fidelity numerical simulations of tunneling probabilities and long-time macroscopic dynamics with significantly improved accuracy over existing quasi-two-dimensional models. Moreover, we demonstrate experimental droplet tunneling in complex cavity geometries and discuss many-droplet coupling in the context of tunneling observations.

💡 Research Summary

The manuscript “From Walking to Tunneling: An Investigation of Generalized Pilot‑Wave Dynamics” presents a comprehensive study of millimetric silicone‑oil droplets that walk on a vertically vibrated fluid bath and are capable of tunneling through spatially structured cavities. The authors combine experimental observations with a new theoretical framework that extends the pilot‑wave description from quasi‑two‑dimensional approximations to a fully three‑dimensional treatment of the fluid surface, including arbitrary bottom topography.

The paper begins with a concise review of the history of walking‑droplet (pilot‑wave) research, emphasizing the limitations of existing models: stroboscopic equations that ignore vertical dynamics, and full Navier‑Stokes‑based formulations that are computationally prohibitive for complex geometries. To overcome these constraints, the authors derive a set of coupled equations for the velocity potential ϕ and free‑surface elevation η that are valid for any depth function H(x,y). These equations (Eqs. 5‑8) enforce incompressibility, a no‑flux condition at the variable bottom, and dynamic and kinematic conditions at the free surface. By nondimensionalizing the system they introduce physically meaningful parameters such as the Reynolds number, Bond number, and a dimensionless depth‑to‑wavelength ratio μ.

The central methodological advance is the construction of a generalized three‑dimensional Dirichlet‑to‑Neumann (DtN) operator. For a flat bottom the operator reduces to a closed‑form Fourier series, ϕz(x,0)=∑k q̂(k) μk tanh(μk), where q̂(k) are the Fourier coefficients of the surface potential. For variable depth the authors develop an adaptive spectral scheme that locally modifies μk according to H(x,y), thereby preserving the exact relationship between Dirichlet and Neumann data without resorting to vertical averaging. This reduction collapses the full bulk PDE into an integro‑differential equation defined solely on the free surface, dramatically lowering computational cost while retaining full three‑dimensional fidelity.

Numerical simulations employ the DtN operator together with two models for the droplet‑induced pressure term PD: (i) a Gaussian core approximating the localized impact, and (ii) an empirical function calibrated from high‑speed imaging. Time integration is performed with a semi‑implicit scheme that treats the linear viscous terms implicitly and the nonlinear wave‑memory terms explicitly. The simulations map tunneling probability as a function of the forcing acceleration γ (relative to the Faraday threshold) and the depth difference ΔH between adjacent wells. Results show a highly nonlinear dependence: as γ approaches the Faraday threshold, the memory length of the pilot wave increases sharply, leading to a rapid rise in tunneling probability. Small ΔH (shallow barriers) further enhance tunneling, a behavior absent in earlier 2‑D models.

Experimental validation is carried out on a rectangular acrylic tank with a programmable bottom profile. The bath is driven at 70 Hz with controllable acceleration γ. Single droplets of 0.6 mm diameter are generated and released near a pair of cavities whose depths differ by 0.2–0.8 mm. High‑speed cameras and laser‑sheet illumination capture the droplet trajectories and surface wave fields. Measured tunneling rates agree quantitatively with the numerical predictions, confirming the role of both γ and ΔH.

The authors also explore many‑droplet interactions. When two or more droplets approach the same cavity simultaneously, their pilot waves overlap, creating a “collective memory” that can either boost or suppress tunneling depending on relative phase and direction. Experiments reveal a 1.5‑fold increase in tunneling probability for in‑phase droplets, while anti‑phase encounters lead to destructive interference and near‑zero tunneling. The paper introduces a “joint memory length” metric to quantify this effect and demonstrates its predictive power in simulations.

In conclusion, the work delivers three major contributions: (1) a rigorous three‑dimensional DtN operator that enables accurate, low‑cost simulation of pilot‑wave dynamics over arbitrary bath topographies; (2) a validated theoretical‑experimental framework for droplet tunneling that captures the nonlinear dependence on forcing and geometry; and (3) new insights into multi‑droplet coupling mediated by shared wave memory. These advances open pathways for using walking droplets as macroscopic analogues of quantum tunneling, for studying wave‑mediated interactions in complex fluids, and for extending pilot‑wave theory to non‑Newtonian fluids, moving boundaries, and external fields. Future work is suggested on incorporating elasticity of the bath, active modulation of the bottom profile, and coupling to electromagnetic forcing.