1932

Abstract

Yves Couder, Emmanuel Fort, and coworkers recently discovered that a millimetric droplet sustained on the surface of a vibrating fluid bath may self-propel through a resonant interaction with its own wave field. This article reviews experimental evidence indicating that the walking droplets exhibit certain features previously thought to be exclusive to the microscopic, quantum realm. It then reviews theoretical descriptions of this hydrodynamic pilot-wave system that yield insight into the origins of its quantum-like behavior. Quantization arises from the dynamic constraint imposed on the droplet by its pilot-wave field, and multimodal statistics appear to be a feature of chaotic pilot-wave dynamics. I attempt to assess the potential and limitations of this hydrodynamic system as a quantum analog. This fluid system is compared to quantum pilot-wave theories, shown to be markedly different from Bohmian mechanics and more closely related to de Broglie's original conception of quantum dynamics, his double-solution theory, and its relatively recent extensions through researchers in stochastic electrodynamics.

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Literature Cited

  1. Anderson JD. 2005. Ludwig Prandtl's boundary layer. Phys. Today 58:45–48 [Google Scholar]
  2. Bacchiagaluppi G, Valentini A. 2009. Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference Cambridge, UK: Cambridge Univ. Press
  3. Bach R, Pope D, Liou S, Batelaan H. 2013. Controlled double-slit electron diffraction. New J. Phys. 15:033018 [Google Scholar]
  4. Bechhoeffer J, Ego V, Manneville S, Johnson B. 1995. An experimental study of the onset of parametrically pumped surface waves in viscous fluids. J. Fluid Mech. 288:325–50 [Google Scholar]
  5. Bell JS. 1966. On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38:447–52 [Google Scholar]
  6. Bell JS. 1982. On the impossible pilot wave. Found. Phys. 12:989–99 [Google Scholar]
  7. Bell JS. 1987. Speakable and Unspeakable in Quantum Mechanics Cambridge, UK: Cambridge Univ. Press
  8. Benjamin TB, Ursell F. 1954. The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225:505–15 [Google Scholar]
  9. Berry MV, Chambers RG, Large MD, Upstill C, Walmsley JC. 1980. Wavefront dislocations in the Aharanov-Bohm effect and its water wave analogue. Eur. J. Phys. 1:154–62 [Google Scholar]
  10. Bohm D. 1952a. A suggested interpretation of the quantum theory in terms of hidden variables, I. Phys. Rev. 85:166–79 [Google Scholar]
  11. Bohm D. 1952b. A suggested interpretation of the quantum theory in terms of hidden variables, II. Phys. Rev. 85:180–93 [Google Scholar]
  12. Bohm D, Vigier JP. 1954. Model of the causal interpretation of quantum theory in terms of a fluid with irregular fluctuations. Phys. Rev. 96:208–16 [Google Scholar]
  13. Bohm D, Hiley BJ. 1982. The de Broglie pilot wave theory and the further development of new insights arising out of it. Found. Phys. 12:1001–16 [Google Scholar]
  14. Bohr N. 1935. Can quantum-mechanical description of physical reality be considered complete?. Phys. Rev. 48:696–702 [Google Scholar]
  15. Boyer TH. 2010. Derivation of the Planck spectrum for relativistic classical scalar radiation from thermal equilibrium in an accelerating frame. Phys. Rev. D 81:105024 [Google Scholar]
  16. Boyer TH. 2011. Any classical description of nature requires classical electromagnetic zero-point radiation. Am. J. Phys. 79:1163–67 [Google Scholar]
  17. Bühler O. 2010. Wave-vortex interactions in fluids and superfluids. Annu. Rev. Fluid Mech. 42:205–28 [Google Scholar]
  18. Burinskii A. 2008. The Dirac-Kerr-Newman electron. Gravit. Cosmol. 14:109–22 [Google Scholar]
  19. Bush JWM. 2010. Quantum mechanics writ large. Proc. Natl. Acad. Sci. USA 107:17455–56 [Google Scholar]
  20. Bush JWM, Oza A, Moláček J. 2014. The wave-induced added mass of walking droplets. J. Fluid Mech. 755: R7. doi: 10.1017/jfm.2014.459 [Google Scholar]
  21. Carmigniani R, Lapointe S, Symon S, McKeon BJ. 2014. Influence of a local change of depth on the behavior of walking oil drops. Exp. Thermal Fluid Sci. 54:237–46 [Google Scholar]
  22. Chebotarev L. 2000. Introduction: the de Broglie-Bohm-Vigier approach in quantum mechanics. Jean-Pierre Vigier and the Stochastic Interpretation of Quantum Mechanics S Jeffers, B Lehnert, N Abramson, L Chebotarev 1–18 Montreal: Apeiron [Google Scholar]
  23. Cole DC, Zhou Y. 2003. Quantum mechanical ground state of hydrogen obtained from classical electrodynamics. Phys. Lett. A 317:14–20 [Google Scholar]
  24. Coste C, Lund F, Umeki M. 1999. Scattering of dislocated wave fronts by vertical vorticity and the Aharonov-Bohm effect. I. Shallow water. Phys. Rev. E 60:4908–16 [Google Scholar]
  25. Couder Y, Fort E. 2006. Single particle diffraction and interference at a macroscopic scale. Phys. Rev. Lett. 97:154101 [Google Scholar]
  26. Couder Y, Fort E. 2012. Probabilities and trajectories in a classical wave-particle duality. J. Phys. Conf. Ser. 361:012001 [Google Scholar]
  27. Couder Y, Fort E, Gautier C-H, Boudaoud A. 2005a. From bouncing to floating: noncoalescence of drops on a fluid bath. Phys. Rev. Lett. 94:177801 [Google Scholar]
  28. Couder Y, Protière S, Fort E, Boudaoud A. 2005b. Walking and orbiting droplets. Nature 437:208 [Google Scholar]
  29. Crommie MF, Lutz CP, Eigler DM. 1993a. Confinement of electrons to quantum corrals on a metal surface. Science 262:218–20 [Google Scholar]
  30. Crommie MF, Lutz CP, Eigler DM. 1993b. Imaging standing waves in a two-dimensional electron gas. Nature 363:524–27 [Google Scholar]
  31. Davisson C, Germer LH. 1927. The scattering of electrons by a single crystal of nickel. Nature 119:558–60 [Google Scholar]
  32. de Broglie L. 1923. Ondes et quanta. C. R. 177:507–10 [Google Scholar]
  33. de Broglie L. 1926. Ondes et mouvements Paris: Gautier-Villars
  34. de Broglie L. 1930. An Introduction to the Study of Wave Mechanics London: Methuen & Co.
  35. de Broglie L. 1956. Une interprétation causale et nonlinéaire de la Mechanique ondulatoire: la théorie de la double solution Paris: Gautier-Villars
  36. de Broglie L. 1964. La thermodynamique cachée des particules. Ann. Inst. Henri Poincaré 1:1–19 [Google Scholar]
  37. de Broglie L. 1987. Interpretation of quantum mechanics by the double solution theory. Ann. Fond. Louis Broglie 12:1–23 [Google Scholar]
  38. de Gennes PG, Brochard-Wyart F, Quéré D. 2002. Gouttes, bulles, perles et ondes Paris: Belin
  39. de la Peña L, Cetto AM. 1996. The Quantum Dice: An Introduction to Stochastic Electrodynamics Dordrecht: Kluwer Acad.
  40. Denardo BC, Puda JJ, Larraza A. 2009. A water wave analog of the Casimir effect. Am. J. Phys. 77:1095–101 [Google Scholar]
  41. Donnelly RJ. 1993. Quantized vortices and turbulence in helium II. Annu. Rev. Fluid Mech. 25:327–71 [Google Scholar]
  42. Dorbolo S, Terwagne D, Vandewalle N, Gilet T. 2008. Resonant and rolling droplets. New J. Phys. 10:113021 [Google Scholar]
  43. Douady S. 1990. Experimental study of the Faraday instability. J. Fluid Mech. 221:383–409 [Google Scholar]
  44. Durr D, Goldstein S, Zanghi N. 2012. Quantum Physics Without Quantum Philosophy New York: Springer
  45. Eddi A, Boudaoud A, Couder Y. 2011a. Oscillating instability in bouncing drop crystals. Europhys. Lett. 94:20004 [Google Scholar]
  46. Eddi A, Decelle A, Fort E, Couder Y. 2009a. Archimedean lattices in the bound states of wave interacting particles. Europhys. Lett. 87:56002 [Google Scholar]
  47. Eddi A, Fort E, Moisy F, Couder Y. 2009b. Unpredictable tunneling of a classical wave-particle association. Phys. Rev. Lett. 102:240401 [Google Scholar]
  48. Eddi A, Moukhtar J, Perrard J, Fort E, Counder Y. 2012. Level splitting at a macroscopic scale. Phys. Rev. Lett. 108:264503 [Google Scholar]
  49. Eddi A, Sultan E, Moukhtar J, Fort E, Rossi M, Couder Y. 2011b. Information stored in Faraday waves: the origin of path memory. J. Fluid Mech. 674:433–63 [Google Scholar]
  50. Eddi A, Terwagne D, Fort E, Couder Y. 2008. Wave propelled ratchets and drifting rafts. Europhys. Lett. 82:44001 [Google Scholar]
  51. Edwards WS, Fauve S. 1994. Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278:123–48 [Google Scholar]
  52. Einstein A, Podolsky B, Rosen N. 1935. Can quantum-mechanical description of physical reality be considered complete?. Phys. Rev. 47:777–80 [Google Scholar]
  53. Everett H. 1957. Relative state formulation of quantum mechanics. Rev. Mod. Phys. 29:454–62 [Google Scholar]
  54. Faraday M. 1831. On the forms and states of fluids on vibrating elastic surfaces. Philos. Trans. R. Soc. Lond. 121:319–40 [Google Scholar]
  55. Feynman RP, Leighton RB, Sands M. 1964. The Feynman Lectures on Physics New York: Addison-Wesley
  56. Fiete GA, Heller EJ. 2003. Theory of quantum corrals and quantum mirages. Rev. Mod. Phys. 75:933–48 [Google Scholar]
  57. Fort E, Couder Y. 2013. Trajectory eigenmodes of an orbiting wave source. Europhys. Lett. 102:16005 [Google Scholar]
  58. Fort E, Eddi A, Boudaoud A, Moukhtar J, Couder Y. 2010. Path-memory induced quantization of classical orbits. Proc. Natl. Acad. Sci. USA 107:17515–20 [Google Scholar]
  59. Gamow G. 1928. The theory of nuclear disintegration. Nature 122:805–7 [Google Scholar]
  60. Gier S, Dorbolo S, Terwagne D, Vandewalle N, Wagner C. 2012. Bouncing of polymeric droplets on liquid interfaces. Phys. Rev. E 86:066314 [Google Scholar]
  61. Gilet T, Bush JWM. 2009a. Chaotic bouncing of a drop on a soap film. Phys. Rev. Lett. 102:014501 [Google Scholar]
  62. Gilet T, Bush JWM. 2009b. The fluid trampoline: droplets bouncing on a soap film. J. Fluid Mech. 625:167–203 [Google Scholar]
  63. Gilet T, Bush JWM. 2012. Droplets bouncing on a wet, inclined surface. Phys. Fluids 24:122103 [Google Scholar]
  64. Gilet T, Terwagne D, Vandewalle N, Dorbolo S. 2008. Dynamics of a bouncing droplet onto a vertically vibrated interface. Phys. Rev. Lett. 100:167802 [Google Scholar]
  65. Gilet T, Vandewalle N, Dorbolo S. 2007. Controlling the partial coalescence of a droplet on a vertically vibrated bath. Phys. Rev. E 76:035302 [Google Scholar]
  66. Goldman DI. 2002. Pattern formation and fluidization in vibrated granular layers, and grain dynamics and jamming in a water fluidized bed. PhD Thesis, Univ. Tex., Austin
  67. Goldstein S. 1987. Stochastic mechanics and quantum theory. J. Stat. Phys. 47:645–67 [Google Scholar]
  68. Grossing G, Fussy S, Mesa Pascasio J, Schwabl H. 2012a. An explanation of interference effects in the double slit experiment: classical trajectories plus ballistic diffusion caused by zero-point fluctuations. Ann. Phys. 327:421–27 [Google Scholar]
  69. Grossing G, Fussy S, Mesa Pascasio J, Schwabl H. 2012b. The quantum as an emergent system. J. Phys. Conf. Ser. 361:021008 [Google Scholar]
  70. Gutzwiller MC. 1990. Chaos in Classical and Quantum Mechanics Berlin: Springer-Verlag
  71. Haisch B, Rueda A. 2000. On the relation between a zero-point-field-induced inertial effect and the Einstein–de Broglie formula. Phys. Rev. A 268:224–27 [Google Scholar]
  72. Haisch B, Rueda A, Dobyns Y. 2001. Inertial mass and the quantum vacuum fields. Phys. Rev. A 268:224–27 [Google Scholar]
  73. Harris DM, Bush JWM. 2013. The pilot-wave dynamics of walking droplets. Phys. Fluids 25:091112 [Google Scholar]
  74. Harris DM, Bush JWM. 2014a. Droplets walking in a rotating frame: from quantized orbits to multimodal statistics. J. Fluid Mech. 739:444–64 [Google Scholar]
  75. Harris DM, Bush JWM. 2014b. Generating uniaxial vibration with an electrodynamic shaker and external air bearing. J. Sound Vib. In press
  76. Harris DM, Moukhtar J, Fort E, Couder Y, Bush JWM. 2013. Wavelike statistics from pilot-wave dynamics in a circular corral. Phys. Rev. E 88:011001 [Google Scholar]
  77. Hestenes D. 1990. The zitterbewegung interpretation of quantum mechanics. Found. Phys. 20:1213–32 [Google Scholar]
  78. Holland PR. 1993. The Quantum Theory of Motion: An Account of the De Broglie–Bohm Causal Interpretation of Quantum Mechanics Cambridge, UK: Cambridge Univ. Press
  79. Jayaratne OW, Mason BJ. 1964. The coalescence and bouncing of water drops at an air-water interface. Proc. R. Soc. Lond. A 280:545–65 [Google Scholar]
  80. Keller J. 1953. Bohm's interpretation of the quantum theory in terms of “hidden” variables. Phys. Rev. 89:1040–41 [Google Scholar]
  81. Kracklauer AF. 1992. An intuitive paradigm for quantum mechanics. Phys. Essays 5:226–34 [Google Scholar]
  82. Kracklauer AF. 1999. Pilot wave steerage: a mechanism and test. Found. Phys. Lett. 12:441–53 [Google Scholar]
  83. Kumar K. 1996. Parametric instability of the interface between two fluids. Proc. R. Soc. Lond. A 452:1113–26 [Google Scholar]
  84. Kumar K, Tuckerman LS. 1994. Parametric instability of the interface between two fluids. J. Fluid Mech. 279:49–68 [Google Scholar]
  85. Labousse M, Perrard S. 2014. Non-Hamiltonian features of a classical pilot-wave dynamics. Phys. Rev. E 90022913
  86. Lieber SI, Hendershott MC, Pattanaporkratana A, Maclennan JE. 2007. Self-organization of bouncing oil drops: two-dimensional lattices and spinning clusters. Phys. Rev. E 75:056308 [Google Scholar]
  87. Lighthill J. 1956. Physics of gas flow at very high speeds. Nature 178:343–45 [Google Scholar]
  88. Madelung E. 1926. Quantentheorie in Hydrodynamischen form. Z. Phys. 40:322–26 [Google Scholar]
  89. Miles J, Henderson D. 1990. Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22:143–65 [Google Scholar]
  90. Moisy F, Rabaud M, Salsac K. 2009. A synthetic Schlieren method for the measurement of the topography of a liquid interface. Exp. Fluids 46:1021–36 [Google Scholar]
  91. Moláček J, Bush JWM. 2012. A quasi-static model of drop impact. Phys. Fluids 24:127103 [Google Scholar]
  92. Moláček J, Bush JWM. 2013a. Droplets bouncing on a vibrating fluid bath. J. Fluid Mech. 727:582–611 [Google Scholar]
  93. Moláček J, Bush JWM. 2013b. Droplets walking on a vibrating fluid bath: towards a hydrodynamic pilot-wave theory. J. Fluid Mech. 727:612–47 [Google Scholar]
  94. Neitzel GP, Dell'Aversana P. 2002. Noncoalescence and nonwetting behavior of liquids. Annu. Rev. Fluid Mech. 34:267–89 [Google Scholar]
  95. Nelson E. 1966. Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150:1079–85 [Google Scholar]
  96. Nelson E. 2012. Review of stochastic mechanics. J. Phys. Conf. Ser. 361:012011 [Google Scholar]
  97. Newton I. 1979. Opticks: Or a Treatise of the Reflections, Refractions, Inflections and Colours of Light Mineola, NY: Dover
  98. Okumura K, Chevy F, Richard D, Quéré D, Clanet C. 2003. Water spring: a model for bouncing drops. Europhys. Lett. 62:237–43 [Google Scholar]
  99. Oza A, Harris DM, Rosales RR, Bush JWM. 2014a. Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization. J. Fluid Mech. 744:404–29 [Google Scholar]
  100. Oza A, Rosales RR, Bush JWM. 2013. A trajectory equation for walking droplets: a hydrodynamic pilot-wave theory. J. Fluid Mech. 737:552–70 [Google Scholar]
  101. Oza A, Wind-Willassen Ø, Harris DM, Rosales RR, Bush JWM. 2014b. Pilot-wave hydrodynamics in a rotating frame: exotic orbits. Phys. Fluids 26:082101 [Google Scholar]
  102. Perrard S, Labousse M, Fort E, Couder Y. 2014a. Chaos driven by interfering memory. Phys. Rev. Lett. 113:104101
  103. Perrard S, Labousse M, Miskin M, Fort E, Couder Y. 2014b. Self-organization into quantized eigenstates of a classical wave-driven particle. Nat. Commun. 5:3219 [Google Scholar]
  104. Philippidis C, Dewdney C, Hiley BJ. 1979. Quantum interference and the quantum potential. Nuovo Cimento 52B:15–28 [Google Scholar]
  105. Pitaevskii L, Stringari S. 2003. Bose-Einstein Condensation New York: Oxford Univ. Press
  106. Prosperetti A, Oguz HN. 1993. The impact of drops on liquid surfaces and the underwater noise of rain. Annu. Rev. Fluid Mech. 25:577–602 [Google Scholar]
  107. Protière S, Bohn S, Couder Y. 2008. Exotic orbits of two interacting wave sources. Phys. Rev. E 78:036204 [Google Scholar]
  108. Protière S, Boudaoud A, Couder Y. 2006. Particle wave association on a fluid interface. J. Fluid Mech. 554:85–108 [Google Scholar]
  109. Protière S, Couder Y, Fort E, Boudaoud A. 2005. The self-organization of capillary wave sources. J. Phys. Condens. Matter 17:S3529–35 [Google Scholar]
  110. Pucci G, Ben Amar M, Couder Y. 2013. Faraday instability in floating liquid lenses: the spontaneous mutual adaptation due to radiation pressure. J. Fluid Mech. 725:402–27 [Google Scholar]
  111. Pucci G, Fort E, Ben Amar M, Couder Y. 2011. Mutual adaptation of a Faraday instability pattern with its flexible boundaries in floating fluid drops. Phys. Rev. Lett. 106:024503 [Google Scholar]
  112. Reynolds O. 1886. On the theory of lubrication. Philos. Trans. R. Soc. Lond. A 177:157–234 [Google Scholar]
  113. Rueda A, Haisch B. 2005. Gravity and the quantum vacuum inertia hypothesis. Ann. Phys. 14:479–98 [Google Scholar]
  114. Schrödinger E. 1930. Uber die kraftefreie Bewegung in der relativistischen Quantenmechanik. Sitz. Preuss. Akad. Wiss. Phys. Math. Kl. 24:418–28 [Google Scholar]
  115. Shirikoff D. 2013. Bouncing droplets on a billiard table. Chaos 23:013115 [Google Scholar]
  116. Spiegel E. 1980. Fluid dynamical form of the linear and nonlinear Schrödinger equations. Physica D 1:236–40 [Google Scholar]
  117. Surdin M. 1974. L'éctrodynamique stochastique et l'interprétation de la Mécanique Quantique. C. R. Acad. Sci. Paris B 278:881–83 [Google Scholar]
  118. Taylor GI. 1909. Interference fringes with feeble light. Proc. Camb. Philos. Soc. 15:114–15 [Google Scholar]
  119. Terwagne D. 2012. Bouncing droplets, the role of deformations PhD Thesis, Univ. Liège
  120. Terwagne D, Gilet T, Vandewalle N, Dorbolo S. 2009. Metastable bouncing droplets. Phys. Fluids 21:054103 [Google Scholar]
  121. Terwagne D, Gilet T, Vandewalle N, Dorbolo S. 2010. From a bouncing compound drop to a double emulsion. Langmuir 26:11680–85 [Google Scholar]
  122. Terwagne D, Ludewig F, Vandewalle N, Dorbolo S. 2013. The role of droplet deformations in the bouncing droplet dynamics. Phys. Fluids 25:122101 [Google Scholar]
  123. Terwagne D, Vandewalle N, Dorbolo S. 2007. Lifetime of a bouncing droplet. Phys. Rev. E 76:056311 [Google Scholar]
  124. Towler M. 2009. De Broglie-Bohm pilot-wave theory and the foundations of quantum mechanics: a graduate lecture course Univ. Camb. http://www.tcm.phy.cam.ac.uk/∼mdt26/pilot_waves.html
  125. von Neumann J. 1932. Mathematische Grundlagen der Quantenmechanik Berlin: Springer
  126. Walker J. 1978. Drops of liquid can be made to float on the liquid. What enables them to do so?. Sci. Am. 238:151–58 [Google Scholar]
  127. Weinberg S. 1995. The Quantum Theory of Fields I Foundations. Cambridge, UK: Cambridge Univ. Press
  128. Weinstein A, Pounder JR. 1945. An electromagnetic analogy in mechanics. Am. Math. Mon. 52:434–38 [Google Scholar]
  129. Wind-Willassen Ø, Moláček J, Harris DM, Bush JWM. 2013. Exotic states of bouncing and walking droplets. Phys. Fluids 25:082002 [Google Scholar]
  130. Yang AL, Chien W, King M, Grosshandler WL. 1997. A simple piezoelectric droplet generator. Exp. Fluids 23:445–47 [Google Scholar]
  131. Yarin AL. 2006. Drop impact dynamics: splashing, spreading, receding, bouncing. Annu. Rev. Fluid Mech. 38:159–92 [Google Scholar]
  132. Young T. 1804. The Bakerian Lecture: experiments and calculations relative to physical optics. Philos. Trans. R. Soc. Lond. 94:1–16 [Google Scholar]
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