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Statistical projection effects in a hydrodynamic pilot-wave system


Millimetric liquid droplets can walk across the surface of a vibrating fluid bath, self-propelled through a resonant interaction with their own guiding or ‘pilot’ wave fields. These walking droplets, or ‘walkers’, exhibit several features previously thought to be peculiar to the microscopic, quantum realm. In particular, walkers confined to circular corrals manifest a wave-like statistical behaviour reminiscent of that of electrons in quantum corrals. Here we demonstrate that localized topological inhomogeneities in an elliptical corral may lead to resonant projection effects in the walker’s statistics similar to those reported in quantum corrals. Specifically, we show that a submerged circular well may drive the walker to excite specific eigenmodes in the bath that result in drastic changes in the particle’s statistical behaviour. The well tends to attract the walker, leading to a local peak in the walker’s position histogram. By placing the well at one of the foci, a mode with maxima near the foci is preferentially excited, leading to a projection effect in the walker’s position histogram towards the empty focus, an effect strongly reminiscent of the quantum mirage. Finally, we demonstrate that the mean pilot-wave field has the same form as the histogram describing the walker’s statistics.

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This work was supported by the US National Science Foundation through grants CMMI-1333242, DMS-1614043 and CMMI-1727565. The authors thank D. Harris and G. Pucci for input.

Author information

P.J.S. and J.W.M.B. conceived and developed the project. P.J.S. and T.C.-P. performed the experiments and reduced the data. P.J.S. and J.W.M.B. wrote the paper.

Correspondence to John W. M. Bush.

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Instantaneous wave field generated by a walking droplet

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Fig. 1: Schematic of the experimental set-up.
Fig. 2: Droplet statistics in an elliptical corral.
Fig. 3: Mean pilot-wave field.
Fig. 4: Fundamental modes of the elliptical corral transiently excited by the walker.
Fig. 5: Resonant projection effects.