Article | Published:

Statistical projection effects in a hydrodynamic pilot-wave system

Nature Physicsvolume 14pages315319 (2018) | Download Citation

Abstract

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.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

  1. 1.

    Couder, Y., Protière, S., Fort, E. & Boudaoud, A. Dynamical phenomena: walking and orbiting droplets. Nature 437, 208–208 (2005).

  2. 2.

    Protière, S., Boudaoud, A. & Couder, Y. Particle–wave association on a fluid interface. J. Fluid Mech. 554, 85–108 (2006).

  3. 3.

    Andersen, A. et al. Double-slit experiment with single wave-driven particles and its relation to quantum mechanics. Phys. Rev. E 92, 013006 (2015).

  4. 4.

    Bush, J. W. M. Pilot-wave hydrodynamics. Annu. Rev. Fluid Mech. 47, 269–292 (2015).

  5. 5.

    Harris, D. M., Moukhtar, J., Fort, E., Couder, Y. & Bush, J. W. M. Wavelike statistics from pilot-wave dynamics in a circular corral. Phys. Rev. E 88, 011001 (2013).

  6. 6.

    Crommie, M. F., Lutz, C. P. & Eigler, D. M. Confinement of electrons to quantum corrals on a metal surface. Science 262, 218–220 (1993).

  7. 7.

    Shirokoff, D. Bouncing droplets on a billiard table. Chaos 23, 013115 (2013).

  8. 8.

    Gilet, T. Dynamics and statistics of wave-particle interactions in a confined geometry. Phys. Rev. E 90, 052917 (2014).

  9. 9.

    Gilet, T. Quantumlike statistics of deterministic wave-particle interactions in a circular cavity. Phys. Rev. E 93, 042202 (2016).

  10. 10.

    Kondo, J. Resistance minimum in dilute magnetic alloys. Prog. Theor. Phys. 32, 37–49 (1964).

  11. 11.

    Fiete, G. A. & Heller, E. J. Theory of quantum corrals and quantum mirages. Rev. Mod. Phys. 75, 933–948 (2002).

  12. 12.

    Moon, C. R., Lutz, C. P. & Manoharan, H. C. Single-atom gating of quantum-state superpositions. Nat. Phys. 4, 454–458 (2008).

  13. 13.

    Manoharan, H. C., Lutz, C. P. & Eigler, D. M. Quantum mirages formed by coherent projection of electronic structure. Nature 403, 512–515 (2000).

  14. 14.

    Harris, D. M. & Bush, J. W. M. Generating uniaxial vibration with an electrodynamic shaker and external air bearing. J. Sound Vibration 334, 255–269 (2015).

  15. 15.

    Harris, D. M., Liu, T. & Bush, J. W. M. A low-cost, precise piezoelectric droplet-on-demand generator. Exp. Fluids 56, 83 (2015).

  16. 16.

    Couder, Y., Fort, E., Gautier, C.-H. & Boudaoud, A. From bouncing to floating: noncoalescence of drops on a fluid bath. Phys. Rev. Lett. 94, 177801 (2005).

  17. 17.

    Douady, S. Experimental study of the Faraday instability. J. Fluid Mech. 221, 383–409 (1990).

  18. 18.

    Faraday, M. On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 299–340 (1831).

  19. 19.

    Miles, J. & Henderson, D. Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22, 143–165 (1990).

  20. 20.

    Eddi, A., Sultan, E., Moukhtar, J., Fort, E., Rossi, M. & Couder, Y. Information stored in Faraday waves: the origin of a path memory. J. Fluid Mech. 674, 433–463 (2011).

  21. 21.

    Moláček, J. & Bush, J. W. M. Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory. J. Fluid Mech. 727, 612–647 (2013).

  22. 22.

    Gutierrez-Vega, J. C., Rodriguez-Dagnino, R. M., Meneses-Nava, M. A. & Chavez-Cerda, S. Mathieu functions, a visual approach. Am. J. Phys. 71, 233–242 (2003).

  23. 23.

    Blanchette, F. Modeling the vertical motion of drops bouncing on a bounded fluid reservoir. Phys. Fluids 28, 032104 (2016).

  24. 24.

    Gluckman, B. J., Arnold, C. B. & Gollub, J. P. Statistical studies of chaotic wave patterns. Phys. Rev. E 51, 1128–1147 (1995).

  25. 25.

    C. Cohen-Tannoudji, B. Diu & F. Laloë. Quantum Mechanics (Wiley, 1977).

  26. 26.

    Kumar, K. Linear theory of Faraday instability in viscous liquids. Proc. R. Soc. Lond. A 452, 1113–1126 (1996).

  27. 27.

    Heller, E. J. Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits. Phys. Rev. Lett. 53, 1515–1518 (1984).

  28. 28.

    Bush, J. W. M. The new wave of pilot-wave theory. Phys. Today 68, 47–53 (2015).

Download references

Acknowledgements

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

Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    • Pedro J. Sáenz
    • , Tudor Cristea-Platon
    •  & John W. M. Bush

Authors

  1. Search for Pedro J. Sáenz in:

  2. Search for Tudor Cristea-Platon in:

  3. Search for John W. M. Bush in:

Contributions

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.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to John W. M. Bush.

Electronic supplementary material

  1. Supplementary Movie

    Instantaneous wave field generated by a walking droplet

About this article

Publication history

Received

Accepted

Published

DOI

https://doi.org/10.1038/s41567-017-0003-x