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

TL;DR

This study addresses 3D droplet tunneling in millimetric walking droplets by developing a generalized hydrodynamic model that couples a three-dimensional velocity potential $\phi$, surface elevation $\eta$, and a moving droplet position $\mathbf{x}_p(t)$ over variable bottom topography $H(\mathbf{x})$. A key advance is the three-dimensional Dirichlet-to-Neumann operator, augmented by Galerkin-derived topographic coefficients $X_{\mathbf{k}}$, which reduces a 3D elliptic problem to tractable 1D ODEs and enables high-fidelity simulations. The authors implement both central-difference and Fourier pseudo-spectral numerical schemes, validate the model against experiments, and demonstrate complex phenomena such as tunneling, including cooperative tunneling, and hydrodynamic angular-momentum-like effects. Experimental results with structured cavities reveal geometry-dependent tunneling probabilities and collective droplet behavior, supporting the framework’s predictive power and potential broader applicability to shallow-water wave dynamics. Overall, the work provides a rigorous, scalable platform for understanding macroscopic pilot-wave dynamics in complex topographies with implications for quantum analogies and coastal-wave modeling.

Abstract

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.

Figures (9)

• Figure 1: A millimetric bouncing droplet and its associated wave field, with the fluid bath bottom topography being a constant-depth circular corral. By setting the oscillatory acceleration $\gamma$ to be $1.6$ g and ensuring that the vibration number $\Omega$ --- which compares the angular forcing frequency to the droplet's oscillatory frequency --- satisfies $\Omega > 0.8$, we ensure that the droplet lies in the simple-bouncing regime of Figure \ref{['fig-regime-diagram']}.
• Figure 2: A phase diagram (adapted from molavcek2013dropsb) at a driving oscillation frequency of 70 Hz, displaying the dependence of droplet dynamics on the forcing parameter $\Gamma = \gamma/\mathrm{g}$ (where $\gamma$ gives the magnitude of oscillatory forcing) and vibration number $\Omega$. The regions denoted by $C$ denote chaos, whereas the regions designated by $(a,b)$, for integers $a,b$, denote $b$ drop contacts occurring within $a$ forcing periods gilet2009chaotic. Our analysis centers on the walking regime, denoted $W.$
• Figure 3: Schematic of the constraints on the elliptic PDE (\ref{['laplacian_1']}) in Theorem \ref{['nondim_procedure']}, which describe droplet dynamics on a fluid bath with variable bottom topography in three spatial dimensions. Equations (\ref{['bottom_1']}–\ref{['kinematical_condition_1']}) describe the conditions for fluid flow along the boundaries of the region, while Equation (\ref{['position_update']}) describes the droplet trajectory itself.
• Figure 4: Workflow for analysis of droplet dynamics. In Figure \ref{['centraldifference']}, we demonstrate central-difference schema workflow, which requires frequent Fourier transforms (FFT and IFFT denote the Fast Fourier Transform and Inverse Fast Fourier Transform, respectively) The Fourier pseudo-spectral method in Figure \ref{['pseudospectral']} provides a more streamlined approach. Created by student researcher.
• Figure 5: A numerical model of the evolution of a droplet wave profile after a single droplet bounce at $(0,0,0)$, seen from a bird's eye view. The impact of a single droplet leads to the formation of surface waves. The wave propagates outward from its initial impact, displaying sharp parallels in overall form and scale to experimental results. The axis scaling is with respect to the Faraday wavelength, $\lambda_F$. Created by student researcher.

Theorems & Definitions (8)

• Theorem 2.1: Navier-Stokes stokes1851effect
• Theorem 2.2: Milewski, et al. milewski2015faraday
• Theorem 3.1
• proof : Proof Sketch
• Theorem 3.2
• proof : Proof.
• Theorem 3.3
• proof : Proof.