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Hydrodynamic quantum analogs
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Reports on Progress in Physics
Rep. Prog. Phys. 84 (2020) 017001 (41pp)

https://doi.org/10.1088/1361-6633/abc22c

Review

Hydrodynamic quantum analogs
John W M Bush1,∗

and Anand U Oza2

1

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, United States ofAmerica2
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, United Statesof America
E-mail: bush@math.mit.edu
Received 29 June 2020, revised 21 September 2020
Accepted for publication 16 October 2020
Published 21 December 2020
Abstract

The walking droplet system discovered by Yves Couder and Emmanuel Fort presents an
example of a vibrating particle self-propelling through a resonant interaction with its own
wave field. It provides a means of visualizing a particle as an excitation of a field, a common
notion in quantum field theory. Moreover, it represents the first macroscopic realization of aform of dynamics proposed for quantum particles by Louis de Broglie in the 1920s. The factthat this hydrodynamic pilot-wave system exhibits many features typically associated with themicroscopic, quantum realm raises a number of intriguing questions. At a minimum, it extends
the range of classical systems to include quantum-like statistics in a number of settings. A
more optimistic stance is that it suggests the manner in which quantum mechanics might becompleted through a theoretical description of particle trajectories. We here review theexperimental studies of the walker system, and the hierarchy of theoretical models developed
to rationalize its behavior. Particular attention is given to enumerating the dynamicalmechanisms responsible for the emergence of robust, structured statistical behavior. Anotherfocus is demonstrating how the temporal nonlocality of the droplet dynamics, as results fromthe persistence of its pilot wave field, may give rise to behavior that appears to be spatiallynonlocal. Finally, we describe recent explorations of a generalized theoretical framework thatprovides a mathematical bridge between the hydrodynamic pilot-wave system and various
realist models of quantum dynamics.
Keywords: walking droplets, pilot-wave dynamics, quantum analogs(Some figures may appear in colour only in the online journal)

the surface of a vibrating liquid bath through a resonant interaction with its own quasi-monochromatic wave field [3–5].
The compound object comprising the droplet and its guiding or ‘pilot’ wave is termed a walker (figure 1). Hydrodynamic quantum analogs is the subject dedicated to exploring
the ability of this hydrodynamic pilot-wave system and related
dynamical systems to exhibit behavior analogous to that arising in quantum systems. The walking-droplet system has led
to hydrodynamic analogs of various microscopic quantum
systems, to varying degrees of success, including tunneling[6–9], the quantum corral [10–14], the quantum mirage [13],Landau levels [15–17], Friedel oscillations [18], diffraction

1. Introduction‘A man may imagine things that are false, but he canonly understand things that are true . . . ’– Sir Isaac Newton
Pilot-wave hydrodynamics is the nascent field of study initiated in 2005 by the discovery of Yves Couder and EmmanuelFort [1, 2] that a millimetric droplet may self-propel along∗
Author to whom any correspondence should be addressed.
‘Corresponding Editor Professor Masud Mansuripur’1361-6633/20/017001+41$33.00

1© 2020 IOP Publishing Ltd

Printed in the UKRep. Prog. Phys. 84 (2020) 017001

Review

Figure 1. Image gallery. (a) A bouncing drop. (b) A walker. (c) Faraday waves apparent just above the Faraday threshold. (d) A droplet
moves unpredictably on a bath forced above the Faraday threshold. (e) The subsurface vortical flow generated by a bouncing drop. (f) A
walking droplet, with its trajectory indicated by the black line. (g) A strobed image of a walker surfing its pilot wave. (h) Well above the
Faraday threshold, Faraday waves break, generating droplets. (a), (d) Reproduced from [159]. CC BY 3.0. (b), (c) Reprinted by permission
from Springer Nature Customer Service Centre GmbH: Journal of Visualization. [161] © 2017. (e) Reprinted figure by permission from [61],Copyright (2014) by the American Physical Society. (f), (g) Reproduced with permission from [4].

only to forces acting instantaneously upon it, while spatial
locality forbids action at a distance. The apparent action at a
distance implied by Coulomb’s law of electrostatics was reconciled through the realization that the forces so defined are
mediated by electromagnetic waves that propagate no fasterthan the speed of light. Pilot-wave hydrodynamics explores
the possibility that the spatial nonlocality of quantum mechanics might likewise be rooted in a currently unresolved wave
dynamics, that quantum non-locality might be a manifestationof spatially local hereditary pilot-wave dynamics.
Particle-wave interactions are common on the macroscopic
scale. Familiar examples include a buoy floating on the surface of the sea, and a grain of sand moving in response toelastic waves induced on a Chladni plate [74]. In both, theparticles move primarily in response to imposed waves rather
than waves of their own creation. Conversely, in the hydrodynamic pilot-wave system, drop motion is driven entirelyby the bouncing drop’s interaction with its own wave field(figure 1). If the droplet were not present, there would be no
pilot wave; if there were no pilot-wave, there would be no lateral drop motion. Identifying the droplet as a self-propellingwave source [40] differentiates the walker system from thegreat majority of classical wave–particle systems.
A key feature of the walking droplet system is that of ‘pathmemory’ [15, 75]. At each impact, the drop responds to thewave field generated by some number of its previous bounces,the precise number being determined by the longevity of its
pilot-wave field. The bath thus serves as the memory of the system, storing the information of the walker’s past, specificallyits path and its environment, in its wave field. The more persistent the waves generated by the bouncing drop, the longer the
memory. The feature of path-memory is apparent in virtually
all of the theoretical models developed to describe the walking droplets [76]. Specifically, there is a propulsive wave force
proportional to the slope of the local wave field whose form
must be deduced by integrating backwards in time in order toaccount for all the waves generated along the walker’s path.
The path memory renders the drop motion non-Markovian, its
dynamics hereditary. Of course, the walker system is local inboth space and time: the instantaneous force acting on the dropfrom slits [2, 19–25], spin states [17, 26, 27], Zeeman splitting[27, 28], the quantum harmonic oscillator [26, 29–35], andthe Hong–Ou–Mandel effect [36]. Assemblages of bouncing
droplets have been shown to exhibit features characteristic of
the microscopic realm, including quantized static and dynamic
bound states [28, 37–50], crystal vibrations [51–53] andspin–spin correlations [54, 55]. Hydrodynamic analogs of various optical systems have also been explored with the walkingdroplet system, including optical waveguides [56], Bragg
scattering [57], optical ratcheting [58] and particle trappingwith the Talbot effect [59, 60]. The walking-droplet system hassuggested classical reinterpretations of a number of traditionally beguiling quantum notions, including wave–particle duality, wave function collapse, superposition of states, statisticalprojection effects, single-particle diffraction and interference,
nonlocality, uncertainty and entanglement.
When successful, hydrodynamic quantum analogs provide
a means of understanding emergent quantum phenomena froma classical, trajectory-based perspective. When unsuccessful,they demarcate the limits of the hydrodynamic system as aquantum analog, and suggest new directions to explore theoretically from a more general pilot-wave framework. The subject thus offers a mechanism for distinguishing between what
can and cannot be understood about quantum systems froma classical perspective. In a field whose most distinguishedpractitioners have been known to glory in the inscrutability of
their subject (for example, Feynman’s dictum ‘Nobody understands quantum mechanics’), this progressive approach should
represent a welcomed change. The implications of the fieldof hydrodynamic quantum analogs for the state of quantum
foundations has been the source of speculation among both
physicists and philosophers [62–71].
Laplacian determinism postulates that, given the initial conditions and governing laws of a physical system, one can predict its evolution indefinitely. This simple state of affairs is lostfor hereditary systems [72, 73], systems with ‘memory’, whose
dynamics are non-local in time, with their evolution depending explicitly on their past. Locality is a feature of a physicaltheory of particle motion if the particle feels only forces acting
locally upon it. Temporal locality requires that a body respond2

Rep. Prog. Phys. 84 (2020) 017001

ReviewMore general walker-boundary interactions, including scattering and diffraction, are treated in section 8. Finally, in section 9
we review the explorations of a generalized pilot-wave framework in which the common features of de Broglie’s quantum
dynamics and the walking-droplet system are retained, but thelimitations and shortcomings of both may be side-stepped. A
summary is presented in section 10, along with projections forfruitful research directions.

is determined by the local form of its pilot-wave, and the pilotwave field evolves in time through a local partial differential
equation. However, the influence of its wave field renders thedroplet dynamics non-local in time: the wave force depends
on the droplet’s history. This temporal non-locality in the drop
dynamics can give rise to behavior that appears to be non-local
in space. Several examples of the possible misinference of spatial nonlocality in the walker system will be reviewed in what
follows.
The other critical ingredient for the emergent quantumlike behavior is resonance between the walker’s bouncing
motion and its wave field. The combination of the bath vibration and this resonance leads to a highly structured, quasimonochromatic wave field that imposes a dynamic constraint
on the walking droplet. The three established paradigms forthe emergence of quantized dynamical states and quantumlike statistics from pilot-wave hydrodynamics, to be detailedin what follows and summarized in section 10.1, are all firmly
rooted in the quasi-monochromatic form of the droplet’s pilot
wave.
The hydrodynamic pilot-wave system is a particular caseof a more general classical pilot-wave dynamics, theoretical
explorations of which have extended the range of quantum
analogs, as reviewed in section 9. Such a generalized pilotwave framework also provides a means of comparing pilotwave hydrodynamics with its quantum predecessors, including
de Broglie’s double-solution pilot-wave theory [68, 77–80],
Bohmian mechanics [81, 82], stochastic dynamics [70, 83,84], and stochastic electrodynamics [85, 86]. It has also leadto the first tentative walker-inspired attempts to formulate a
trajectory-based quantum mechanics [20, 70, 87–89].
Before proceeding, it is appropriate to define the limited
scope of this review. Hydrodynamics has been a rich sourceof metaphor and physical analogy for microscopic physics[4], from Newton’s description of photons as stones skipping through the ether [90] to Lord Kelvin’s description of
microscopic particles as vortical knots in the ether [91]. The
walking-droplet system has a significant number of precursors
in terms of hydrodynamic quantum analogs. These include
Thomas Young’s use of ripple-tank experiments to demonstrate the wave nature of light [92], hydrodynamic analogs of
the Aharanov–Bohm [93] and Casimir [94] effects, and a classification of droplet vibration in terms of a dynamic periodic
table [95]. The specific goal of this report is to review work
done on the walking-droplet system since its discovery, and tooutline its intriguing connections to existing realist models ofquantum dynamics.
By way of providing both motivation and historical context,
in section 2 we discuss the quantum predecessors of the hydrodynamic pilot-wave system. In section 3, we provide a brief
overview of the walking-droplet system, and in section 4 detailthe hierarchy of theoretical models developed to describe it. In
section 5, we review experimental and theoretical studies ofstatic and dynamic bound states formed from the interactionsof multiple bouncing droplets. We devote section 6 to a review
of orbital pilot-wave dynamics, as may arise when the walking droplet is subjected to an external force. In section 7, we
review studies of walker motion in a confined corral geometry.

2. Quantum pilot-wave theories
Quantum mechanics is a theory that describes the statistical behavior of microscopic particles. The absence of a satisfactory description of an underlying dynamics has left thefield in a state comparable to that of thermodynamics prior
to the development of statistical mechanics [96]. Moreover, it
has left the door open to the proliferation of quantum interpretations, attempts to comprehend what the theory actuallymeans [97–99]. Such interpretations have run the gamut frommystical to perfectly sensible. According to the Copenhagen
interpretation, the statistical description provided by quantum mechanics is complete: there is no underlying physicalreality, or notion of quantum particles following trajectories.
The insistence on the completeness of the statistical description of quantum systems leads immediately to various difficulties, including superluminal wave-function collapse [100],the confounding notion of complementarity, and the paradox of Schrödinger’s cat. The philosophical extravagance of
the Copenhagen interpretation opened the door to equallyuntestable rivals, notable among them being the Many-Worlds
interpretation [101], according to which the universe branches
into copies of itself in response to any decoherence event suchas the act of human observation.
Following the preeminence of the Copenhagen interpretation, a number of relatively commonsensical perspectives
have emerged, and are evidently gaining ground. According to
the statistical [102] or ensemble interpretation [103], quantum
mechanics describes the statistical evolution of an ensemble ofsimilarly prepared states. Consistent histories [104] dispenses
with many of the difficulties of other interpretations, including wave function collapse and non-locality, and ensures aframework in which classical rules of probability are satisfied
by formulating a self-consistent history of the system evolution. While undoubtedly improvements over their comparatively immoderate predecessors, these perspectives remain but
interpretations, providing no physical picture as to what form
the underlying quantum dynamics might take.
Virtually all attempts to envisage a realist quantum dynamics have involved particles interacting with a background field,
either a coherent pilot wave as proposed by de Broglie [77–79]
and Bohm [81, 82], or a stochastic vacuum field as positedin stochastic dynamics [83, 84] and stochastic electrodynamics [85, 86]. As we shall see, experimental and theoreticalmodeling of pilot-wave hydrodynamics has, in various parameter regimes, captured certain features of all such theories.
Since these theories of quantum dynamics have yet to be satisfactorily completed, they remain classified as interpretations.
However, it is worth noting that, if successful in providing a3

Rep. Prog. Phys. 84 (2020) 017001

Review

dynamical picture that is consistent with the statistical predictions of standard quantum mechanics in all settings, they will
leave no room for interpretation.

of the second yields the evolution equation for the quantum
velocity of probability v q :m2.1. For whom the Bell does not toll

(3)

From the perspective of the fluid mechanician, theMadelung formulation [113] has two attractive features [114].
First, it consigns Planck’s constant  to a single term, the coefficient on the quantum potential. Second, the system corresponds to that describing the evolution of a shallow, inviscid
fluid layer if one associates ρq with the fluid depth, and v q withthe depth-averaged fluid velocity. When Q is linearized about astate of uniform ρq , it assumes the form of the surface-tensioninduced curvature pressure in shallow-water hydrodynamics[115]. One thus sees the first tentative relation between theroles of  in quantum statistics and surface tension σ in hydrodynamics, a relation that has been deepened by a comparison
between the walker system and de Broglie’s physical picture[3] (see section 2.4).
The hydrodynamic formulation that follows from the
Madelung transformation is the standard approach to describing a number of quantum systems, including Bose–Einsteincondensates [116]. As it represents no more than a mathematical reformulation of the Schrödinger equation, it adds no
new physics, only a means of interpreting the evolution of
the statistics of a quantum system in terms of fluid flow. The
walking-droplet system suggests a much richer hydrodynamic
interpretation of quantum mechanics, a physical picture repletewith the notion of particles generating waves and moving in
response to them [3–5], a dynamics evocative of de Broglie’s
original pilot-wave theory (see section 2.4).

It is widely believed that the work of John Bell, specificallythe derivation of Bell’s inequality [105] and the experimental
violation thereof [106–109], sounds the death knell for hidden
variable theories. A more defensible inference is that experimental violation of Bell’s inequality requires that any hidden
variable theory be nonlocal in the quantum sense [110]. Morgan [111] and Vervoort [66, 67] assert that Bell’s inequalities
have no bearing on hidden-variable theories in which particles interact through a background wave field. Bell himself,
who derived his inequality, then saw it violated by the experiments of Aspect [106–108], came to the conclusion that there
must be a pilot-wave dynamics underlying quantum statistics[110, 112]. Specifically, he supported the de Broglie–Bohm
theory, according to which particles interact with a nonlocal
quantum potential [112]. Readers who believe that they understand Bell’s work better than Bell himself are invited to revisit
his original writings [110], along with a recent account of thesystematic misinterpretation thereof [98].
Any theory, hidden variable or otherwise, must answer tothe same experimental data. All features of quantum mechanics must be captured by a successful hidden variable theory,
including quantum non-locality. In the absence of a hidden
variable theory, the quantum non-locality inferred by the quantum correlations measured experimentally [106–108] requires
superluminal signaling, ‘spooky action at a distance.’ A successful theory would provide a rational explanation for suchcorrelations based on the particle dynamics, thereby obviating
the need for superluminal signaling.

2.3. Bohmian mechanics

Bohmian mechanics [81, 82] was formulated by David Bohm
in 1952 in an attempt to reintroduce the notion of particle trajectory to quantum mechanics. It is briefly reviewed here inorder to distinguish it from the relatively rich double-solution
pilot-wave theory of de Broglie, which the walking droplet
system more closely resembles [3, 117]. Unlike de Broglie’s
mechanics, Bohmian mechanics was not rooted in relativity: its starting point was the linear Schrödinger equation (1).
The basic assertion of Bohmian mechanics is that the particle
velocity ẋp is identical to the quantum velocity of probability v q ; consequently, the trajectory of the particle is prescribedby equation (3). The particle thus moves in response to theclassical and quantum potentials. Notably, Bohmian mechanics posits no mechanism for wave generation. The only wave
in Bohmian mechanics is the wavefunction of the standard
quantum formulation, as necessarily satisfies Schrödinger’s
equation (1) and prescribes the statistical behavior of thesystem.
Bohmian mechanics immediately drew criticism from several circles, a common objection being to the nonlocal formof the quantum potential [118]. In response to these criticisms,
Bohm and Vigier [119] invoked a stochastic background field.
They then viewed the Bohmian trajectories as streamlines in agas flow, but imagined that the real particles would stray from

2.2. The Madelung transformation

The original hydrodynamic interpretation of quantum mechanics follows directly from the Madelung transformation [113],
which allows the linear Schrödinger equation (LSE) to berecast in hydrodynamic form. Consider the standard quantum
mechanical description of a particle of mass m in the presence of a potential V (x, t). The particle’s wave function Ψ(x, t)
evolves according to the LSE:


2 2
(1)
iΨt = − ∇ + V Ψ,
2mwhere  is the reduced Planck’s constant. Expressing thewave function in polar form, Ψ(x, t) = R(x, t) eiS(x,t)/ , andsubstituting into (1) yields
Dρq+ ρq ∇ · v q = 0,
Dt

Dv q= −∇Q − ∇V.
Dt∂S 1 2+ mv q + Q + V = 0, (2)∂t2where v q (x, t) = ∇S(x, t)/m is the quantum velocity of probability, D/Dt = ∂t + v q · ∇ the material derivative, S(x, t) theaction, ρq (x, t) = R2 the probability density, and Q(x, t) =
2
 1
− 2m
R ∇ R the quantum potential. The first equation in (2)expresses the conservation of probability. Taking the gradient2

4

Rep. Prog. Phys. 84 (2020) 017001

Review

oscillators exchanging their rest mass energy m0 c2 with fieldenergy ω at the Compton frequency. In a manner left unspecified by de Broglie, the particle vibration generates a pilot
wave with the de Broglie wavelength λB that guides the particle. The synchrony of the particle’s internal oscillation with
its pilot wave in all reference frames, as follows from relativistic considerations [68, 88], was referred to by de Broglie
as the ‘harmony of phases’, and considered to be a key feature of his pilot-wave mechanics, ‘une grand loi de la nature’[134]. Notably, synchrony between the droplet and its pilot
wave is also critical to the emergent quantum-like behavior of
the hydrodynamic pilot-wave system.
de Broglie envisioned a quantum particle as having aninternal vibration, with the Compton frequency, that creates a pilot-wave field satisfying the Klein–Gordon equation[130, 135],(4)
φtt − c2 Δφ + ωc2 φ = 0.

these streamlines in response to stochastic fluctuations, just as
dust particles in an air flow stray from streamlines in responseto Brownian motion [3, 117].
Particle diffraction from slits is treated in Bohmian mechanics by solving Schrödinger’s equation for the wavefunction,from which the form of the quantum potential is calculated.
The Bohmian particle paths are then computed via (3) using
an ensemble of initial conditions corresponding to a Gaussianbeam. Philippidis [120] thus deduced a self-consistent result,
specifically, particle paths consistent with the wavefunctionused to derive them. Recent advances in weak measurement
have reported that mean particle paths in the single and double
slit experiments are consistent with the predictions of Bohmianmechanics [121]. Given that Bohmian mechanics can be seen
as a dynamical reformulation of a statistical theory, it would
seem optimistic to imagine that it might describe anything
more than a mean dynamics. Of course, a description of themean dynamics is an improvement over no description at all;thus, Bohmian mechanics has found wide application in both
optics [122] and physical chemistry [123, 124]. Moreover, as
noted by Bell [110, 112], workers in Bohmian mechanics [71,96, 125–128] have provided a valuable perspective that has
acted to counter the prevailing complacency surrounding themeaning of quantum mechanics.
The walking droplet system demonstrates how quantumlike statistics may, in certain circumstances, emerge fromclassical pilot-wave dynamics. It is significantly richer thanBohmian mechanics, replete with a mechanism for particleinduced wave generation, and with a real, particle-centeredwave to complement the statistical wave form. While theseadded complexities of the walker system make it a more challenging dynamical system to analyze, it allows one to see themanner in which one might circumvent the difficulties inherent
in Bohmian mechanics, namely the need to invoke a nonlocal quantum potential and a stochastic background field. The
manner in which the hydrodynamic pilot-wave system does
so is evocative of a predecessor of Bohmian mechanics, deBroglie’s double-solution pilot-wave theory.

The particle thus excites a field of monochromatic waves
and moves in response to the gradient of the wave phase(see table 3). The de Broglie wavelength, λB = 2π/kB , represents the wavelength of the monochromatic wave form
whose group speed corresponds to the particle speed:dω(k ) = v. The de Broglie relation p = γ L m0 v = kB , where
dk B
−1/2

γL = 1 − (v/c)2is the Lorentz factor, follows directlyfrom the dispersion relation of the Klein–Gordon equation:
ω 2 = ωc2 + k2 c2 . The resulting physical picture is that of aparticle being propelled by a monochromatic wave form thatdresses the particle, an image manifest in the walking droplet
system.
While there are similarities between de Broglie’s conception and the hydrodynamic pilot-wave system, there are also
a number of qualitative differences. First and foremost, deBroglie did not specify the manner in which the particle vibration generates the wave: equation (4) is unforced. Second, theKlein–Gordon equation is undamped, while the waves accompanying the walking drop are damped by viscous effects.
Third, de Broglie’s trajectory equation posits that the particle
responds to gradients in the phase of its pilot-wave field: particle inertia plays no role. As we shall see in what follows, thereis accumulating evidence that the low-drop-inertia, undamped
wave limit is that in which the quantum-like behavior of thewalking droplets is most pronounced.
De Broglie’s greatest obstacle was identifying the form of
his pilot wave φ. While he asserted that the particle vibration
was responsible for wave generation, he did not express mathematically the manner in which this generation arose. He statedthat the pilot wave satisfied the Klein–Gordon equation (4),but did not demonstrate the manner in which a particle vibrating at the Compton frequency would generate such a waves,or the form that it would take. Several possibilities were thus
explored. First, he assumed that φ was monochromatic, fromwhich p = kB follows directly from his trajectory equation(table 3). Following the work of Bohm [81, 82], he proposedthat φ took the same form as the Ψ wave of equation (1), apartfrom a nonlinearity in the vicinity of the particle [80, 130].
With this concession, Ψ ∝ φ, de Broglie’s double-solution
theory effectively merges with Bohmian mechanics, hence the2.4. de Broglie’s matter waves

de Broglie’s double-solution pilot-wave program [77–80, 129,130] was an attempt to reconcile quantum mechanics with relativity [68]. His basic premise was that there are two thingsin the Universe, light and matter. Since light was then known
to have both wave and particle natures, he proposed that so
too must matter, hence de Broglie’s matter waves. He positedthat particles have an internal vibration, and move in concert
with their matter waves. On the basis of his physical picture, deBroglie predicted electron diffraction, the experimental confirmation of which [131, 132] earned him the Nobel Prize inphysics in 1929.
Relativity requires that a particle of rest mass m0 haveenergy E = m0 c2 , while quantum mechanics requires thatE = ω. Equating these two equations, referred to by Wilczek[133] as a ‘poem in two lines’, yields the Einstein–de Broglie
relation, m0 c2 = ω, as defines the Compton frequency, ω c =m0 c2 /. According to de Broglie, quantum particles are5

Rep. Prog. Phys. 84 (2020) 017001

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so-called de Broglie–Bohm pilot-wave theory, which is effectively Bohmian mechanics [136]. The hydrodynamic pilotwave system presents an example where the instantaneous
pilot wave differs substantially from the emergent statistical form [10, 13], which is more in line with de Broglie’s
original conception [3, 117]. A recent attempt to extend de
Broglie’s double-solution mathematical program, informedby the walking-droplet system [88, 89], will be described in
section 9.3.
de Broglie presented an early form of his theory at theSolvay conference of 1927, but drew criticism from several circles, and his theory was not thereafter widely accepted [137].
Indeed, de Broglie himself abandoned it for a time before
returning to it following the work of Bohm [81, 82]. Nevertheless, it provided a number of the cornerstones of quantum theory. In addition to inspiring the de Broglie relation,p = kB , and the Einstein–de Broglie relation, m0 c2 = ω, his
physical picture suggested that many of the quantum mysteries may be rooted in an unresolved dynamics on the Compton
scale, the scale of particle vibration [138]. His physical picture inspired Schrödinger to develop both the Klein–Gordon
and Schrödinger equations in order to describe the pilot wave
[139]. Subsequently, these wave equations were adopted todescribe the wavefunction, and so only the statistical behavior of quantum systems, a development to which Schrödinger
himself objected [140]. The physical picture that had inspired
the development of the principal governing equations of quantum mechanics was thus abandoned in favor of . . . no physicalpicture.
Compelling as de Broglie’s double-solution theory may
seem in light of a number of the alternatives, it was never completed. In particular, because the generation mechanism andform of the pilot-wave field φ were not specified, it was notpossible to demonstrate that the proposed pilot-wave dynamics would give rise to statistical behavior described by theΨ wave. Furthermore, the physical origin of the pilot wave
field was not specified. In his later works, de Broglie [80]sought its origins in a stochastic background field, the modern equivalent of which would be the quantum vacuum [141].
While the electromagnetic vacuum field has been most thoroughly investigated as a potential pilot-wave field [85, 86],
other possibilities exist. Notably, the wave equation (4) considered by de Broglie is precisely that describing the Higgs
field [88].
Before proceeding, it is worth enumerating the many lines
of evidence supporting de Broglie’s proposal of associating
a quantum particle with a high frequency oscillation at theCompton frequency ω c . First, it is required from dimensional
analysis if one assumes that the relevant fundamental constants are , c and m0 . Second, it provides the means by whichto connect solutions φ of the Klein–Gordon equation (4) tosolutions Ψ of the linear Schrödinger equation (1). Specifically, if φ(x, t) = Ψ(x, t)eiωc t is a solution of the Klein–Gordon
equation, then Ψ(x, t) satisfies the LSE provided the particle speed v  c [142]. Zitterbewegung was a feature of earlymodels of quantum theory taken to indicate a high-frequency
jitter of microscopic particles at ω c [143]. The classical modelof the electron is that of a charge orbiting the Compton radius,

λc = h/(mc) at the Compton frequency [144]. Particle vibration at ωc plays a critical role in both Hestenes’ Zitterbewegunginterpretation of quantum mechanics [145] and in stochastic electrodynamics [85, 86]. Notably, Compton frequencies
are not yet observable experimentally for most particles; forexample, the Zitter frequency of the electron is 1.6 × 1021 s−1 .
Nevertheless, some evidence of Zitterbewegung has recently
been reported in Bose–Einstein condensates [146] and trapped
ion systems [147].
2.5. Stochastic dynamics

While the pilot-wave theories of de Broglie and Bohm rely
on a particle being guided by a coherent wave form, eitherthe Ψ wave in Bohm’s mechanics or the unspecified φ wave
in de Broglie’s, stochastic dynamics posits that microscopic
particles are propelled by a stochastic background field associated with the quantum vacuum. Nelson [83, 84] introduced thestochastic interpretation of quantum mechanics. Starting withthe linear Schrödinger equation (1), he argued that the evolution of the quantum statistics of a microscopic particle of mass
m may be understood in terms of a diffusive process characterized by diffusivity /(2m). If one asserts that a microscopic
particle has momentum p = kB , then this diffusivity may bereexpressed as D ∼ vλB , as one expects to arise from a random walk of a particle with mean speed v and step length λB .
As will be seen in what follows, similar random walks havebeen found in a number of theoretical models of pilot-wavehydrodynamics.
A modern extension of de Broglie’s double-solution
pilot-wave theory has emerged from the field of stochastic
electrodynamics, according to which the pilot wave is electromagnetic in origin [86]. A number of lines of evidence
suggest that, even at zero Kelvin, there is electromagnetic
energy in the quantum vacuum field [141]. Consistency withthe requirement of Lorentz invariance and experiments on the
Casimir effect indicates that this zero-point field has a particular spectral form: the vacuum mode with frequency ω hasenergy U(ω) = ω/2 [148]. Notably, the quantum vacuum
thus provides a natural means by which to introduce Planck’s
constant  into a classical theory. The successes and limitations of stochastic electrodynamics in rationalizing quantum
phenomena have been recently reviewed by Boyer [149].
In de la Peña and Cetto’s conception of stochastic electrodynamics [85, 86], the stochastic background field excites
the vibration of the particle at its natural frequency. The particle then interacts with resonant vacuum modes: the resulting
pilot-wave is thus an electromagnetic wave with the Compton
frequency and Compton wavelength in the particle frame ofreference. The de Broglie wave arises due to the particle
motion [86], and represents the envelope of this Dopplershifted, Lorentz-transformed pilot wave [85]. As posited,stochastic electrodynamics provides alternative rationale for anumber of quantum phenomena [150–152], but its ability tocapture diffraction from slits has been questioned [153, 154].
The relation between stochastic dynamics and stochastic electrodynamics has recently been reviewed by de la Peña et al[155].
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Figure 2. Regime diagram indicating the dependenceof the drop’s bouncing behavior on the dimensionless vibrational acceleration of thebath, γ/γ F , and vibration number, Ω = ω/ σ/ρa3 , for a bath with viscosity 20 cS driven at ω/(2π) = 80 Hz. Drop dynamics both below[156] and above [157] the Faraday threshold, γ/γ F = 1, are shown. Reprinted from [157], with the permission of AIP Publishing.

3. The walking droplet system

resulting waves; specifically, the Faraday waves always havean oscillation frequency that is an integer multiple of ω/2.
In all cases considered to date in the context of pilot-wavehydrodynamics, the most unstable mode is subharmonic, withhalf the frequency of the bath’s vibrational forcing. The wavelength λF of the resulting Faraday wave field is then prescribedby the water-wave dispersion relation, which relates the Faraday wavenumber kF = 2π/λF to the frequency of vibrational
forcing ω and the wave frequency ω w :


σk3
(5)
(ω/2)2 = ωw2 (kF ) ≡ gkF + F tanh kF H.
ρThe hydrodynamic pilot-wave system consists of a millimetricdrop placed on the surface of a vibrating liquid bath (figure 1).
The walking droplet system is visually striking, its aesthetic
appeal evident even with a simple classroom demonstration[158–161]. Quantitative experiments are typically performed
on a fluid bath of silicone oil with viscosity between 20 and
100 cS vibrating at 20–100 Hz. While all are based on theoriginal design of Couder and Fort’s experiments [2, 38, 162],
a number of refinements have improved precision and repeatability [25, 49, 163–165]. The drop behavior is known todepend critically on both drop size and driving acceleration[38, 41, 156, 166–168]; consequently, the generation of reproducible data requires precise control of both. Based on thedrop-generator designs of Terwagne et al [169] and Yang et al[170], Harris et al [165, 171] thus developed a device to produce uniform droplets with diameters ranging from 0.5 to 1.4
mm (accurate to ±1%). A number of experiments have indicated the importance of isolating the walkers from air currents
with a lid [13, 23]. This requirement has prompted the development of launchers that allow for the experiments to run continuously, even with a lid [18, 23, 45, 46, 172]. Most recently,
Ellegaard and Levinsen [25] achieved temperature control ofthe bath, thereby eliminating spurious effects associated withdrift of the ambient room temperature.

The form taken by the Faraday wave field may be influenced
by the bounding geometry for small domains, but a square
planform is the norm in sufficiently large domains.
Beyond the Faraday instability threshold, γ > γ F , the Faraday wave system has a number of curious features, including
some common to quantum systems and quantum chaos [177].
In certain parameter regimes, the most unstable wave pattern
is a mixed state consisting of two distinct wavelengths, theresult being a quasi-crystalline wave structure [176, 178]. In
the chaotic wave regime arising for γ  γ F , ‘scars’ emerge[179, 180], corresponding to regions of anomalous wave
energy. Scars are also prominent features of quantum billiards,
which describe the wavefunction of a particle confined to adomain and colliding perfectly elastically against the domain’s
boundary [181]. Finally, there is a critical vibrational acceleration γ R ∼ (σ/ρ)1/3 ω 4/3 above which the vibration prompts
interfacial fracture: surface tension is incapable of stabilizing the wave, whose crests pinch off into drops with size
comparable to λF (figure 1(h)) [182, 183].

3.1. Faraday waves

Consider a horizontal liquid bath of depth H, density ρ, surface tension σ and kinematic viscosity ν subjected to a vertical vibration of amplitude A0 , frequency ω and acceleration
γ sin ωt (where γ = A0 ω2 ), in the presence of a gravitational
acceleration g. Faraday waves [173] develop on the surface
when a critical vibrational acceleration, the Faraday threshold
γ F , is exceeded [174–176] (figure 1(c)). The system vibration has the feature of naturally imposing a quantization on the

3.2. Bouncing and walking drops

A fluid drop of radius a, density ρ, mass m = 4πa3 ρ/3and surface tension
σ is an oscillator with a natural fre
quency ωd ∼ σ/m prescribed by a balance between fluid
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Figure 3. Vertical dynamics of bouncing and walking droplets. (a) Kymographs, deduced by juxtaposing a vertical array of pixels throughthe droplet’s centerline, illustrate the drop’s bouncing [167] and walking [156] modes: in order, (1, 1) bouncing, (2, 2) bouncing, (2, 1)2
walking and chaotic walking. (b) Dependence of a bouncing droplet’s touch-down (red) and take-off (blue) times on the bath’s vibrational
acceleration γ/γ F . The kinematic-match model of the droplet’s vertical dynamics (curves) [193] (see section 4.4) is compared withexperiments (shaded regions) [194]. The vertical lines indicate the transitions between different bouncing modes in experiment (solid) andsimulations (dashed). (c) The dependence of the impact phase Φ (black lines) and horizontal speed u0 (blue line) on the vibrational forcing
γ/γ F , as predicted by the theoretical model of Milewski et al [44] (see section 4.4). The droplet speed u0 is nondimensionalized by the phasespeed, Cp , of a wave with the Faraday wavelength. Background colors denote different bouncing and walking modes. The impact phase
necessarily takes multiple values in regimes with more than one impact per bouncing cycle. The abrupt jump in both phase and speed
indicates the transition between the (2, 1)1 and (2, 1)2 walking modes. (a) Reproduced with permission from [156, 167]. © Cambridge
University Press. (b) Adapted with permission from [193]. © Cambridge University Press. (c) Reproduced with permission from [44].
© Cambridge University Press.

in the vertical dynamics of both bouncing and walking states.
In the bouncing mode (i, j) p, a droplet bounces j times over
i vibration periods [167, 192]. The subscript p orders different modes with the same bouncing periodicity according totheir mechanical energies, with large p denoting bouncers withlarger bouncing amplitude.
Consider a drop of radius a = 0.4 mm, and viscosityν = 20 cS forced at ω/(2π) = 80 Hz, corresponding to a horizontal line in figure 2 with vibration number Ω = 0.87. Just
above the coalescence threshold, the drop bounces with thesame period as the bath vibration, in a (1, 1) mode (figure 3(a)).
The amplitude of the bouncing droplet then generally increases
progressively with driving acceleration and the impact phaseshifts until the period of the bouncing reaches twice that ofthe vibrational forcing, and so becomes commensurate withthe subharmonic Faraday waves favored by the bath vibration.
Figures 3(b) and (c) illustrate the evolution of the bouncing
dynamics during this progression. Drops in the resulting resonant (2, 1) bouncing states may have one of two impact phases
relative to the bath vibration, and so be either in-phase or outof-phase with respect to other droplets. The transition to theresonant (2, 1) state coincides with a substantial increase in the
bouncer’s wave energy, and is soon followed by the onset ofhorizontal droplet motion, the transformation from bouncing
to walking.
In the walking state, a droplet self-propels in a straight line
through a resonant interaction with its own wave field. A variety of walking states exist. Resonant (2, 1) walkers arise when
the bouncing frequency matches the Faraday frequency. In
the parameter regime reported in figure 2, two different resonant walking states are possible, characterized by different
impact phase and denoted by (i, j) = (2, 1)1 or (2, 1)2 [156].
In the (2, 1)1 and (2, 1)2 modes, the drop is in contact withthe bath for approximately T F /2 and T F /4, respectively, where
T F = 4π/ω is the Faraday period [38, 44, 50, 156, 167, 195]

inertia and curvature forces induced by surface tension[184, 185]. When a millimetric fluid drop is placed on
the surface of a vibrating bath, it may be levitated on thebath surface, either hovering or bouncing [186]. The criterion for non-coalescence is that the air layer between bathand drop maintain a critical thickness (of order 100 nm)
during the course of each impact; otherwise, van der Waals
forces acting between the two liquid phases initiate coalescence [162, 187, 188]. When the drop bounces, its distortion into an oblate spheroid during impact increases thesurface area of the underlying air film, thereby discouraging
coalescence. When the drop bounces at its natural frequency,
there is a resonance in the system. Hydrodynamic quantum
analogs arise when there is an additional resonance, between
the bouncing and the resulting subthreshold Faraday waves.
Because the Faraday waves are viscously damped, the system
may then be viewed as a damped oscillator forced at its resonant frequency. For a bath of a given viscosity, the longevity of
the waves generated by the droplet impact, as determines thesystem’s ‘path-memory’, increases with the vibrational acceleration. The terms ‘memory’ and vibrational acceleration arethus used interchangeably henceforth.
The evolution of the bouncing droplet system with increasing vibrational acceleration at a fixed frequency has been characterized experimentally in a number of studies [37, 38, 156,
167, 168, 189]. The system behavior depends on the chosen
fluid, as well as the vibrational frequency. The most extensive walking regime arises with the 20 cS–80 Hz combination [41, 156, 168], the behavior of which is summarized infigure 2. The horizontal axis indicates the magnitude of thevibrational acceleration, while the verticalaxis indicates thevibration number, Ω = ω/ωd = ω/ σ/ρa3 , which indicatesthe relative magnitudes of the bath’s vibrational frequency
and the drop’s natural frequency, and serves as a proxy fordrop size [167, 190, 191]. Different periodicities are evident
8Rep. Prog. Phys. 84 (2020) 017001

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acceleration approaches the Faraday threshold, γ → γ F , effectively erasing its wave field in the process. Sampara and Gilet[200] examined the behavior of the walker system driven withtwo commensurate frequencies. A number of novel behaviors were reported, including chaotic walking near the Faraday
threshold marked by a stop-and-start dynamics. Valani et al[201] recently showed that such a two-frequency forcing may
give rise to a new class of walkers dubbed ‘superwalkers’,
which are twice the size of the traditional walkers and selfpropel at up to four times the speed owing to their relatively
prolonged impact with the bath and the concomitant lateral
acceleration induced by impact (figures 5(c) and (d)). The
superwalkers open the door to a new parameter regime forpilot-wave hydrodynamics and invite further explorations of
the role of multiple-frequency vibrational driving.

(figures 3(a) and (b)). In the high-memory limit, γ → γF− ,this resonance may be lost, and the bouncing may becomechaotic; nevertheless, the drop walks in a straight line [168].
The range of drop sizes over which walking arises for a particular fluid and vibrational frequency is limited. For example,
in figure 2, the smallest walkers have a radius of 0.25 mm andthe largest, 0.47 mm. It is thus, presumably, that this hydrodynamic pilot-wave system was not discovered until 2005
[1].
A striking feature of the walking droplets is the stabilityof the free walking state; indeed, in the absence of interactions
with boundaries or other walkers, the walkers self-propel alonga straight line. This stability is all the more beguiling when one
considers that the drop is riding on the crest of its pilot wave
field (see figures 1(b) and (g)). The simplest intuitive argument
suggests that such an arrangement should be unstable. If thedrop is perturbed backwards, it will receive a relatively small
kick from its wave field at impact, so slow down; if perturbed
forward, it will receive a relatively large kick, so speed up.
What this reasoning fails to take into account is the stabilizing
influence of the system memory. When the drop decelerates oraccelerates, the effective density of wave sources along its path[196] increases or decreases, respectively, thereby increasingor decreasing the amplitude of the local wave field in such a
way as to stabilize the walking state. The robust stability of
the free walking state should thus remind us not only of theimportance of memory in pilot-wave hydrodynamics, but of
the subtlety of hereditary systems.
Tambasco and Bush [157] explored the behavior of thewalker system just above the Faraday threshold, and reported anumber of new dynamical states, characterized by zig-zagging
and meandering motion, as well as a diffusive regime arising at high memory. Higher resonances in the drop oscillation may also be excited. Gilet et al [190] and Dorbolo et al[191] examined the bouncing of relatively large oil drops ofradii a ∼ 0.75 mm and viscosity ν < 100 cS on a highly viscous bath (ν = 1000 cS). Levitation criteria were deduced, andnonaxisymmetric modes of droplet vibration excited, leading
to drop propulsion via tumbling. Hubert et al [197] examined the dependence of the bouncing threshold on the forcing
frequency with a similar arrangement. They found that thebouncing threshold exhibits a minimum when the vibration
number Ω ≈ 1 and a maximum when Ω ≈ 3, which correspond respectively to the drop’s principal modes of deformation acting in resonance and antiresonance with the bathvibration. Most recently, it has been shown that bouncing
droplets may be achieved with liquid metals, specifically, analloy of gallium and indium [198]. While walking states have
yet to be achieved with these liquids, this discovery may further extend the dynamical range of hydrodynamic pilot-wave
systems.
Novel phenomena have also been observed by modulating either the phase or frequency of the vibrational forcing. Perrard et al [199] showed that, when a walker is subjected to an abrupt increase in the forcing acceleration, it may
exhibit a π-shift in its bouncing phase, and so encounter a perfectly reversed wave field at impact. It thus reverses direction,
retracing its trajectory for a time that increases as the forcing

3.3. Pilot-wave field measurements

The novelty of the walking-droplet system lies in the subtle
interplay between the droplet and its guiding wave. It thus
behooves us to understand the precise form of the wave field
generated by the drop impact on a free surface. The pilotwave field in the walking droplet experiment has a characteristic amplitude of 1–20 microns and wavelength λF ∼ 4.8 mm(figure 4). Its duration on the free surface can be as long as
several seconds in the high-memory limit (γ → γ F ). Visualization with a semi-reflective mirror [175] allows one to see
its form. Moisy et al [202] developed a robust means of quantitative measurement of the pilot wave field, the free-surface
synthetic Schlieren technique, which deduces the surface
topography from the apparent distortion of a backgroundimage. The resulting quantitative measurements of the wavefield have been used to characterize walker interactions withboundaries [24, 172, 203] and to benchmark both theoreticalmodels and numerical simulations [75, 204].
Each time the drop impacts the surface, it generates
a transient wave front with a characteristic amplitude of
1–20 microns that propagates radially with a speed of 5–20 cm
s−1 [44, 75, 204]. Behind this front, a quasi-monochromaticwave form persists with a wavelength λF prescribed by thesurface wave dispersion equation (5). The superposition ofmany such waves yields the wave field of a stationary resonant bouncer, as illustrated in figure 4(a). While slow outward
propagation of the zeros of this wave field indicate a weak traveling component [44, 204], the waves may in many instances
be adequately described in terms of a quasi-monochromaticstanding Faraday wave field [75]. The form of the walker wave
field is illustrated in figure 4(b). Experimental measurements
and simulations both indicate its characteristic horseshoe-likeform [38, 44, 75, 204, 205]. As the walker is a moving sourceof waves, the Doppler effect is evident in its wavefield, theapparent wavelength decreasing and increasing, respectively,
up- and downstream of the walker [75].
Tadrist et al [49] brought unprecedented experimental precision to bear on the walker system. Specifically, they developed an experimental technique for inferring the precise timeand location of each drop impact by visualizing the outward
propagating transient wave front generated at impact. Their
9

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Figure 4. Bouncer and walker wave fields (a = 0.38 mm, f = 80 Hz). (a) Measured wave field of a period-doubled bouncer (γ/γ F = 0.77)[204]. The gray curve shows a projection of the cross-section of the wave field passing through the droplet’s center. The gray cylinder
indicates the droplet’s projected shadow. (b) Comparison between experimental measurements (top) and simulation (bottom) of the walker
wave field (γ/γ F = 0.966) [204]. The wave fields of (c) bouncing (γ/γ F = 0.73) and (d) walking (γ/γ F = 0.9) drops, as predicted byMilewski et al [44]. Color bars indicate the wave height in microns. Scale bars denote λF  4.8 mm. In panels (b) and (d), the walkers moveto the right. (a), (b) Reprinted by permission from Springer Nature Customer Service GmbH: Experiments in Fluids. [194] © 2016. (c),(d) Reproduced with permission from [44]. © Cambridge University Press.

technique allows for resolution of the fast vertical dynamics
responsible for wave generation, the uncertainty in which they
identified as a source of chaos in the walker system [49].
Early studies of pilot-wave hydrodynamics were undertaken in relatively deep fluid layers, for which the Faraday wavelength λF  2πH, which requires bath depths of
H > 5 mm for the range of frequencies (50–80 Hz) typicallyconsidered. The resulting Faraday waves then decay before
reaching the lower boundary, so the bottom topography has anegligible influence on the walker. Recent studies have shownthat drops may also achieve a robust walking state in shallow
water; specifically, Sáenz et al [13, 18] demonstrated that drops
may walk in layers as thin as 1.6 mm. Chu et al [61] and Tsaiet al [206] visualized the vortical flows beneath relatively large(a ∼ 1 mm) bouncing droplets, and proposed that asymmetry in these vortices, induced either by a partner droplet or aweakly sloping bottom topography, could lead to propulsion.
Finally, in many hydrodynamic quantum analogs, the boundaries of the fluid bath are generally marked by shallow ‘beach’regions of typical depth 0.1 mm. While this geometry serves tolocally damp the drop’s pilot wave, it introduces uncertainty inthe form of the interfacial boundary condition to be applied intheoretical models, which is necessarily neither Dirichlet (zeroamplitude) nor Neumann (zero-slope) [11, 12, 21, 22, 56, 204,
207–209].

description of inviscid Faraday waves, which was subsequently built upon to incorporate the role of fluid viscosity[211, 212]. Müller et al [213] presented a model for Faraday
waves valid in the weak dissipation limit ν/ωλ2F  1, wherein
the viscous boundary layers are small relative to the Faraday
wavelength. Through a systematic asymptotic expansion of thelinear water-wave dispersion relation with viscous corrections,they obtained an integro-differential Mathieu-type equation
for the time evolution of the wave height. These models of
Faraday waves have provided valuable touchstones for thedevelopment of the hierarchy of theoretical models describing resonant walkers, specifically, the motion of a droplet in
response to the sub-threshold Faraday pilot-wave field that it
generates by bouncing at the Faraday frequency. The models
for resonant walkers can be roughly divided into two classes,
those that approximate the wave field as the sum of monochromatic waves (sections 4.1–4.3), and those that solve the relevant partial differential equations for the evolution of the freesurface (sections 4.4–4.5).
Because the Weber number characterizing the droplet
motion is typically small, We = ρvz2 a/σ ∼ 0.1, with v z being
the droplet’s impact speed, the drops are only weakly perturbed by impact on the surface. Moreover, since the intrusion
depth of the drop into the bath is small, the hydrostatic pressures generated during this impingement are negligible relative to the surface-tension-induced curvature pressures. Thus,
the dominant physics of drop impact on a bath is captured
by considering drop impact on a soap film, a problem examined experimentally and theoretically by Gilet and Bush [192,214]. They demonstrated that, provided the film deflection

4. Theoretical modeling
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break down as γ approaches γ F . Moláček and Bush [167]experimentally characterized the dependence of the droplet’scoefficient of restitution and contact time with the bath on
the Weber number, We = ρż2 a/σ, and the resulting empirical
relations allowed them to express c1 and c2 in terms of We.
Due to discrepancies between the experimentally measured
bouncing mode transitions and those predicted by equation (7)at high memory, Moláček and Bush [167] developed a logarithmic spring model to account for the effect of bath anddrop deformations on the vertical dynamics. Their model
successfully rationalized the regime diagrams that detail thedependence of the bouncing behavior on the dimensionlessvibrational acceleration and vibration number [156, 167, 168](see figure 2).
To describe the walking state, Moláček and Bush [156]derived a trajectory equation for the droplet’s horizontal displacement xp (t):is sufficiently small, the soap film acts on the drop like alinear spring with a spring constant proportional to the surface tension σ. Such linear spring models also satisfactorily
describe the vertical motion of a stationary drop bouncing
in place on a liquid bath at low memory [167]. However, athigh memory the bouncing state is affected by the drop’s relicwave field [50, 156]. Describing the emergent walking state
requires a description of the horizontal force balance, which
in turn requires a model for the pilot-wave form generated bysuccessive impacts.
4.1. Path-memory model

Emmanuel Fort developed the first theoretical model of walking droplets, a discrete-time model for a period-doubled resonant walker [2, 15, 26, 42]. The drop is modeled as a point
particle and the impacts as instantaneous, occurring at times
tn = nT F . The vertical dynamics is assumed to be decoupledfrom the horizontal. The walker is subject to two forces in thehorizontal direction: a linear drag force, and a propulsive force
proportional to the gradient of the wave height h̃ at the walker’s
position. The walker’s horizontal velocity is updated at each
impact according to these two forces, and the horizontal position evolved accordingly. The model thus assumes the formof an iterated map in the walker’s horizontal position xn andvelocity v n at time t = tn :xn+1 = xn + v n+1 TF ,v n+1 = e−Dts /m v n − |vz |∇h̃(xn , tn ) .

mẍp + D(t) ẋp = −F(t)∇h(xp , t).

(8)Here, F(t) = m(z̈ + g(t)) is the normal force exerted by thedrop on the bath, and D(t) = 6πa0 μa + C ρa/σF(t) is thetime-dependent drag coefficient, which accounts for both the
aerodynamic drag imparted during flight and the momentum
transfer from drop to bath during impact. The constant C wasshown to lie in the range 0.17  C  0.33 in the parameter
regime of interest, using measurements of the walker’s coefficient of tangential restitution. The form of the wave field h(x,t)
was deduced using a quasi-potential formulation for the evolution of the free surface, assuming that the bath is forced near
the Faraday threshold, γ  γ F , and that the walker’s impact on
the bath is point-like in both space and time:(6)

Approximate values for the drop’s vertical velocity v z , dragcoefficient D and contact time ts are deduced from experiments. The form of the wave field taken in their path-memory
model has been successively refined over time [26, 75, 196].
Most recently, the time-averaged wave height h̃ is given by
time-averaging equation (9) (to√ be developed in section 4.2)
over the impact time, with the t-term removed for the sake ofanalytical convenience [26]. This model has successfully rationalized the emergent quantum behavior in a number of settings[15, 26, 29, 42, 117, 215, 216] to be detailed in sections 5–8.

t/TF

√

h(x, t) =
n=−∞

× cos

ÃS
J0 (kF |x − xp (tn )|)t − nTF

ωt −(t−tn )/TM
e
.
2

(9)

Here, S is one of two parameters that characterize the droplet’s
impact phase:S=

4.2. Pilot-wave hydrodynamic models

TF
0 F(τ ) sin(ωτ /2) dτ,TF
0 F(τ ) dτC=

TF
0 F(τ ) cos(ωτ /2) dτ.
TF
0 F(τ ) dτ(10)
Moláček and Bush [156] derived an expression for Ã in termsof the fluid parameters, and showed that the memory time
T M is related to the forcing acceleration γ through TM =Td (1 − γ/γF )−1 , where Td ≈ (2kF2 ν)−1 is the viscous decay
time of waves in the absence of vibrational forcing, a scaling
derived previously by Eddi et al [75] using a multiple-scale
perturbation expansion. The dimensionless memory parameter M e = T M /T F prescribes the number of prior bounces that
influence the walker [75]. In typical experiments, M e is in therange 3–400, and quantum-like features tend to emerge for
M e > 10. The fact that M e diverges as γ → γ F simply reflects
the shortcomings of the linear wave approximation in thislimit, where the waves are expected to become nonlinear: thesystem behavior is not singular at the Faraday threshold, but
the linear wave models used to describe it are.

Moláček and Bush [156, 167] developed a hydrodynamicallyconsistent model for bouncer and walker motion based on a
reduction of the Navier–Stokes equations. They demonstrated
that the droplet’s vertical position z(t) on a vibrating fluid bath
may be described to leading order in terms of a linear springmodel:(7)mz̈ + H(−z) (c1 ż + c2 z) = −mg(t),where g(t) = g + γ sin ωt is the effective gravity in the vibrating bath frame of reference and H is the Heaviside function, that ensures that the bath-induced spring force acts only
during impact. Here, z = 0 at the point where the drop’s
lowermost extremity impacts the unperturbed bath surface.
Implicit in this model is the assumption that the drop impacts
an unperturbed interface, that the waves generated by priorimpacts have decayed to zero, an assumption known to11

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Note that the walker’s vertical and horizontal dynamics arecoupled in equation (8), owing to the dependence of F(t) andD(t) on the walker’s vertical position z(t). Predictions madeon the basis of the trajectory equation (8) and wave model (9)exhibited good agreement with experimental data on the walking threshold, and the dependence of the walking speed ondrop size and forcing acceleration. Moreover, these equations
form the basis for the stroboscopic models to be discussed in
section 4.3. The primary shortcoming of the model is that thephase parameters S and C were not resolved, so were instead
combined into a single fitting parameter SC constrained byexperiments to lie in the range 0.1–0.3 [50].
An improvement to the wave form (9) was derived by
Tadrist et al [217], whose developments were based on Müller
et al’s [213] linear Faraday wave model:

drop is assumed to be in resonance with its Faraday wavefield, bouncing at the subharmonic frequency ω F = ω/2. Oza
et al [205] approximated the sum in equation (9) by an integral, an approximation valid provided the timescale of thewalker’s horizontal motion is long relative to the bouncing
period, λF /| ẋp |  TF . Time-averaging equation (9) over thebouncing period then yields the trajectory equation
 t
2mgASC∇
J0 (kF |x − xp (s)|)
mẍp + D ẋp = −
TF
−∞
 
−(t−s)/TM
ds 
,
(12)
×e
x= xp (t)where D = Cmg ρa/σ + 6πaμa is the drag coefficient aver√aged over the Faraday period and A = Ã/2 TF . Note thatthe t−1/2 -temporal decay evident in equation (9) was removed
from the wave kernel for the sake of analytical expediency, on
the grounds that it is subdominant to the exponential decay
term.
Equation (12) represents an integro-differential equation
for the walker’s horizontal trajectory. The product of phaseparameters SC is again taken to be a constant in the range
0.1–0.3, chosen to best match experimental data on the freewalking speed [156]. The simplicity of the stroboscopic model(12) makes it amenable to mathematical analysis and numerical simulation. The authors showed that a bouncer transitions
to a walker via a supercritical pitchfork bifurcation as the forcing acceleration is progressively increased. They concluded
that the walking state is stable to in-line perturbations and neutrally stable to lateral perturbations in the parameter regime
accessible in the laboratory. The stroboscopic model has beenused to study the walker’s complex nonlinear dynamics in a
number of settings [17, 27, 31, 32, 34, 36, 45, 46, 48, 205,
219–221] to be detailed in sections 5–8.

t/TF

h(x, t) =ÃSωt√
J0 (kF |x − xp (tn )|) cos
t
−
nT
2
F
n=−∞
× exp −t − tnα|x − xp (tn )|2−,t − tnTM(11)where 1/α ≈ 2c2g /νkF2 has the units of diffusivity and cg isthe group velocity of waves with wavenumber kF . This wave
form incorporates a Gaussian kernel that accounts for spatial damping, the inclusion of which brings predictions for the
standing wavefield generated by a bouncer in line with experimental measurements [204]. Equation (11) indicates that the
wavefield of a bouncer
decays exponentially over a character
istic lengthscale TM /α [204]; thus, the spatial and temporal
decay rates of sub-threshold Faraday waves are simply related.
Tadrist et al [217] also calculated the wave field generated bya rectilinear walker, and found that it exhibited a Doppler shiftsimilar to that reported in experiments [38, 75], a feature notcaptured by the waveform (9). Moreover, they found that the
decay length δ of the wavefield in the wake of a rectilinear
walker has an angular dependence, δ → (4αu0 (1 + cos θ))−1
in the limit γ → γ F , where u0 is the walking speed and θ theangle with respect to the walking direction.
The bouncing drop models of Terwagne et al [218] andBlanchette [209] also incorporate the deformation of thedroplet, and reproduce many features of the bouncing drop,
including the period-doubling transition between bouncing
modes as the vibrational acceleration is increased progressively. In a recent study of droplet pairs, Couchman et al [50]found that when the influence of the relic surface waves generated by prior impacts is considered, the linear spring model(7) is preferable to its logarithmic counterpart [156]: not only
can it be solved analytically, but it improves the agreement
between theory and experiment. They thus reverted to the simpler linear spring model (7), but explicitly modeled the influence of the relic surface waves on the drop’s vertical dynamics.
Their model will be further discussed in section 5.2.

4.4. Partial differential equations for the fluid interface

The wave generated by a single impact is dominated by the
Faraday wavelength, which corresponds to the slowest decaying Faraday wave mode [156, 167]. In the models described sofar, the wave field has thus been expressed as a sum (9) or integral (12) of monochromatic waveforms generated along thewalker’s path. A more sophisticated class of theoretical models
expresses the wave height as the solution to a partial differential equation describing the free surface evolution. While thecapillary waves generated on the surface of a quiescent bathby both moving [222–224] and oscillating [225] point sources
have been characterized theoretically, the sub-threshold waves
generated by a small source oscillating in resonance with avibrating liquid bath have only been studied recently, in thecontext of pilot-wave hydrodynamics.
Milewski et al [44] developed a quasi-potential wave model
consisting of partial differential equations for the velocity
potential φ(x, z, t) and wave height h(x, t). They considered
the linearized Navier–Stokes equations for an incompressible,
weakly viscous fluid, and employed the formulation of Lamb[226] and Dias et al [227], who derived viscous corrections
to the free surface boundary conditions by accounting for the

4.3. The stroboscopic model

In the stroboscopic model of Oza et al [205], the drop’s vertical dynamics is eliminated from consideration: the walker
is modeled as a continuous source of standing waves. The
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is irrotational and quiescent at infinity, the velocity potential
satisfies
for z < 0, φ → 0 as z → −∞,(Δ + ∂zz ) φ = 0

dynamics and instead assumed the walker to execute instanpositions
taneous point-like impacts at times tn = nT F and 
xn = xp (tn ), so the applied pressure is PD = mgT F n δ(x −xn )δ(t − tn ). This simplification allowed the wave model tobe solved semi-analytically, through a decomposition of the
wave field into the eigenmodes of the Laplacian, Φm (x; k)= J m (kr) cos mθ and Ψm (x; k) = J m (kr) sin mθ:(13)where Δ = ∂xx + ∂yy is the 2D Laplacian. The boundary conditions applied on the interface z = 0 are∞

ht = φz + 2νΔh,φt = −g(t)h +am (t; k)Φm (x; k) + bm (t; k)Ψm (x; k) k dk.
m=0

0

1
σ
Δh + 2νΔφ − PD (x − xp (t), t). (14)
ρ
ρ

(16)
The authors showed that the mode amplitudes am and bm bothsatisfy the Mathieu equation for t = tn ,äm + 4νk2 ȧm + 4ν 2 k4 + ωw2 (k) + kγ sin(ωt + β) am = 0,

The pressure force PD exerted by the drop on the bath isassumed to be nonzero and uniform over an area of radius
R(t):
F(t)
,
PD (x, t) =πR(t)2∞h(x, t) =(17)
where F(t) = max(mz̈ + mg(t), 0) (15)where β is the phase difference between the drop bouncing andbath vibration. During flight, the drop’s inertia is balanced bya drag force −D ẋp and any applied force F:

and R(t) is related to the drop size. The drop’s vertical positionz(t) is assumed to evolve according to the logarithmic springmodel of Moláček and Bush [156]. The model predictions fora bouncer’s wave field compare well with experimental measurements [204]. The model captures both the Doppler shift in
the walker wavefield [38, 75] and the transient wave propagating radially outwards following impact [75, 204], features not
captured by the purely monochromatic wave models discussed
in sections 4.1–4.3.
Galeano-Rios et al [193, 195] considered the impact ofa small non-wetting rigid sphere on a quiescent bath, anddemonstrated that the contact force between the sphere andbath emerges naturally from a kinematic match condition
between the bath and sphere surface. They employed the wave
model (13) and (14), but replaced the ad hoc pressure forcing with two kinematic constraints. First, the free surface was
required to conform to the sphere surface beneath it. Second,
the free surface was required to be tangent to the sphere surface
at their line of convergence. These conditions are sufficient todetermine the air pressure beneath the sphere, without the needto assume a specific form for either the spatial pressure distribution or the contact area, as was required in equation (15).
Their framework was used to model bouncing drops, walkers and superwalkers, and lead them to conclude that drop
deformation plays a significant role only for superwalkers[201]. While the model exhibited good agreement with thebouncing regime diagram reported by Wind-Willassen et al[168], it predicts the onset of chaotic bouncing at vibrational
accelerations slightly below those reported in experiments,
and wave amplitudes slightly larger than those reported byDamiano et al [204]. The authors attribute these relatively
minor shortcomings to the limitations of the quasi-potential
wave approximation (13) and (14).

where [ f (tn )]+− denotes the jump in f at time t n , and W m = π
for m > 0 and W 0 = 2π.
Using fundamental matrices to evolve the system between
successive impacts, the authors recast equations (16)–(20) as
an iterated map that evolves the wave field and walker position
between impacts. By neglecting the details of the drop-bath
impact, this formulation is more computationally efficient than
Milewski et al’s [44] model (equations (13) and (14)) and so
better suited to simulating the walker’s long-time dynamics.
It is also analytically tractable, permitting stability analysis ofboth bouncing and walking states. The energy of a walker’s
wave field was shown to be less than that of a bouncer provided γ > γ W , which suggests an energetic rationale for thetransition from bouncing to walking [33].

4.5. Discrete-time model

4.6. Low-memory limits of the trajectory equation

Durey and Milewski [33] employed the wave model inequations (13) and (14) to simulate the dynamics of a resonant
walker. They neglected consideration of the walker’s vertical

The theoretical models for the walker dynamics may be simplified substantially in the low-memory limit. Protière et al [38]proposed a model that only considered the influence of a single

mẍp = −D ẋp + F(xp ),t = tn .

(18)

The drop impact results in a force being applied to the bath,consideration of which yields jump conditions on am andbm . Similarly, the drop receives a propulsive force from thebath during impact, −mgT F δ(t − tn )∇h(xn , tn ), which yields ajump condition on its velocity. These jump conditions may be
expressed as[ȧm (tn ; k)]+− = −mgkF TFΦm (xn ; k),
ρWm


+
mgkF TFḃm (tn ; k)=−Ψm (xn ; k),
(19)
−
ρWm
  mg+∇h(xp (tn ), tn ) + ẋp (tn− ) ,ẋp (tn ) − = − 1 − e−DTF /m
D
(20)

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Reviewprior impact, and used it to qualitatively capture the dynamics
of single and interacting walkers near the walking threshold.
Filoux et al [43] and Rahman [228] used similar models to
examine the dynamics of trains of walkers in a circular annulus(see section 5.1).
Starting from the stroboscopic model (section 4.3), Bushet al [229] derived a trajectory equation valid in the asymptotic
limit of weak acceleration, |ẍp |TM /| ẋp |  1:d(mw (v)v) + Dw (v)v = F.
dtboundary condition in equation (14) was approximated by
ht = −∇ · (b(x)∇φ) + 2νΔh,(22)where b(x) = tanh(k(x)H(x))/k(x), and k(x) satisfies thewater-wave dispersion relation (5) in each region. Changes
in topography were thus modeled through changes in thelocal phase speed of the Faraday waves through the function b(x). As in Durey and Milewski [33], the pressure forcing in equation (14) was assumed to be point-like in both
space and time. Faria’s model [22] has been used successfully to rationalize a number of walker-boundary interactions
detailed in section 8, including wall reflection [203], diffraction by slits [23], and scattering by submerged pillars [172] andwells [18].

(21)

Here mw (v) and Dw (v) prescribe, respectively, the dependenceof the walker’s effective mass and drag force on its speed
v = | ẋp |, and F is an externally imposed force. The nonlinear
drag force acts always to restore the walker to its free walking
speed, u0 . Since mw (v) > m, the wavefield effectively imparts
an added mass to the walker, an effect responsible for the
anomalously large circular orbits observed in a rotating frame[15, 16] and harmonic potential [29, 220] in the low-memory
regime (see section 6). Equation (21) is referred to as the ‘boost
model’, as it indicates the role of the pilot-wave in increasing
the walker mass relative to the droplet mass by a boost factor
mw (v)/m > 1, a development inspired by those in stochastic
electrodynamics suggesting that interaction with the electromagnetic quantum vacuum may alter a microscopic particle’s
inertial mass [230, 231]. Labousse and Perrard [232] proposed
a Rayleigh oscillator-type model for the walker
 dynamics inwhich Dw (v) is proportional to (v/u0 )2 − 1 , which may beseen as a special case of equation (21) valid in the low-speed(λF /u0  T M ), low-memory limit [233].

4.8. Summary

The theoretical models for the dynamics of free walking
droplets described in this section are listed in table 1. Somehave been as faithful as possible to the hydrodynamic system;others have simply captured its essential features. Still others,
developed to elucidate specific walker behavior, will be discussed in what follows. They can be categorized according
to whether they account for the walker’s vertical dynamics.
Models that neglect the vertical dynamics [33, 205, 235] aremore amenable to mathematical analysis and more numerically tractable, making them well-suited to characterizing awalker’s long-time dynamics and emergent statistical behavior. However, these models have the impact phase as a freeparameter, and do not satisfactorily capture the interactions
between walkers. Conversely, models that consider the fullvertical dynamics [44, 156, 193, 195] capture variations inbouncing phase but are computationally intensive. Couchman
et al’s [50] variable-phase model (see section 5.2) plays anintermediate role by accounting for slow variations in thewalker’s vertical dynamics while remaining mathematicallyand computationally tractable [76].

4.7. Modeling boundary interactions

A number of important hydrodynamic quantum analogs
involve the interaction of walkers with submerged topography or boundaries. To address this class of problems, a number of theoretical models have been developed since that of
Couder and Fort [2, 117]. Nachbin et al [8] introduced a modelallowing for consideration of one-dimensional walker motion
above a two-dimensional fluid region in the (x, z)-plane withdiscontinuous bottom topography z = −H(x). The authors
built upon Milewski et al’s [44] quasi-potential wave model(section 4.4). Specifically, they augmented equations (13) and(14) with the no-penetration boundary condition ∂φ/∂n = 0
on z = −H(x), n being the unit normal to the boundary, andused conformal mapping to solve numerically the resulting
elliptic boundary-value problem. The model has been used toinvestigate both unpredictable walker tunneling (section 8.1)
and the role of wave-mediated coupling in inducing distantwalker–walker correlations [234]. The approach of Nachbin
et al [8] was extended to two dimensions by Durey et al[207] using a different computational method based on domain
decomposition.
Faria [22] extended Milewski et al’s [44] quasi-potential
model to simulate the motion of a resonant walker above a bathwith a piecewise-linear depth profile H(x), specifically, deepand shallow regions separated by discrete steps. The kinematic

5. Multiple droplet interactions
Bouncing droplets may interact through their mutual wave
field to form a variety of bound states, either stationary or
dynamic (figure 5). These bound states invariably exhibit aform of quantization; specifically, there are a finite number ofpreferred arrangements whose relative stability is prescribedby the system parameters. In addition to providing insight into
the manner in which classical resonant wave sources interact [38, 40, 236], characterizing these bound states has leadto progressive refinement of the theoretical models of pilotwave hydrodynamics. Specifically, examining their stability
has elucidated the influence on the drop dynamics of spatial damping and the propagating transient on the pilot-wave
field, and the effects of variations in the vertical bouncing[45–47, 50].
Protière et al [38] considered the collisions of two walking droplets, demonstrating that the pair may either scatteror lock into an orbiting pair [1] (figure 5(a)). For the case of
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Table 1. Summary of theoretical models of walker dynamics. Computational times are given as orders of magnitude of time taken to simulate

the motion of a single (2, 1) droplet on a desktop computer (with a 2.7 GHz Intel Core i5 processor), relative to real time in the experiments.
Model and authors

Vertical dynamics

Wave model

Comp. time/real timePath-memory model(Fort et al [15])Uncoupled fromhorizontal dynamics

Bessel functions generated
by point impacts

O(1)Pilot-wave hydrodynamic model(Moláček and Bush [156])

Logarithmic spring

Stroboscopic model(Oza et al [205])

Constant bouncing
phase

Bessel functions generated
continuously along trajectory

Variable bouncing phase(Couchman et al [50])Bessel functions withspatial damping [50, 76, 217]

No vertical dynamics

Wave effects enter
through nonlinear dragRayleigh oscillator model(Labousse and Perrard [232])
Boost model(Bush et al [229])

O(10−3 )Wave effects enter throughnonlinear drag, added mass
Quasi-potential,
weakly viscousO(102 )

Faraday pilot-wave
model (Milewski et al [44])Logarithmic spring

Discrete Faraday pilot-wave model(Durey and Milewski [33])Discrete,
instantaneous impacts

O(10−1 )Kinematic match pilot-wave model(Rios et al [193, 195])

Kinematic match ofdrop and bath surface

O(104 )

drops of different size, more complex orbital motions arose,
including wobbling circular orbits and epicycles [40, 236]. The
problem of walker collisions was revisited experimentally andtheoretically by Tadrist et al [49]. By resolving the fast dynamics associated with the droplet bouncing, they concluded that,in the high-memory regime, the outcome of a walker collision cannot be predicted simply on the basis of the horizontal
impact geometry. Rather, the system is chaotic, the origins of
the unpredictability being uncertainty in the vertical bouncing
dynamics. The validity of this conclusion in other settings iscurrently being explored.

model (12) with that in (11), thereby substantially improving
the agreement between theory and experiment.
Borghesi et al [42] presented a combined experimental andtheoretical study of promenading pairs, the dynamic boundstate arising when two droplets walk side by side with theirseparation distance either constant or oscillating periodicallyin time (figure 5(b)). They reported quantized interdrop distances, and demonstrated that such pairs move slower thantheir free counterparts. The system behavior was describedtheoretically in terms of a wave interaction energy. Arbelaiz
et al [46] revisited the promenade mode in order to characterize the dependence of the stability of these dynamic boundstates on the system memory.
In their studies of orbiting and promenading walker pairs,
respectively, Oza et al [45] and Arbelaiz et al [46] foundthat a straightforward generalization of the stroboscopic model
to multiple walkers failed to capture the detailed stability
characteristics of these bound states. Specifically, the model
predicted the onset of instability at forcing accelerations considerably below those observed experimentally. The source of
this discrepancy was deduced to be the assumption of constant impact phase SC; thus, the authors inferred an empiricalfunctional form for the dependence of the phase on the forcing
acceleration γ and the local wave height at the walker’s position. The resulting stroboscopic model with phase adaptation
thus indirectly accounted for slow variations in the walker’s
vertical dynamics without explicitly modeling them. Inclusion of this ad hoc model for phase adaptation substantially
improved the match between theory and experiment. Together,
these studies motivated the development of the variable-phase5.1. Dynamic bound states: orbiting, promenading andratcheting pairs, droplet strings

For identical drops, Protière et al [38] showed that circular orbits with quantized orbital radii arose, with different
sets of radii accessible to pairs bouncing in-phase and outof-phase. The orbital quantization was rationalized in termsof the assumed undulatory form of the pilot wave field. They
also noted that the speed of the orbiting pairs is generally lessthan that of a free walker: the smaller the orbits, the slower
the drops. The problem of identical orbiting droplet pairs wasrevisited by Oza et al [45], who characterized the influenceof memory on the stability of the quantized orbital states. By
categorizing the possible orbital states and the dependence of
their stability on memory, their study prompted the refinementof the stroboscopic model, making clear the importance of spatial damping of the pilot wave field on the stability of orbiting
pairs. The authors incorporated the effects of spatial damping[217] by replacing the wave kernel J 0 (kF r) in the stroboscopic
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Figure 5. Bound states of bouncing droplets. (a) An orbiting pair [161]. (b) A promenading pair [159]. A superwalking pair in (c) chasingand (d) orbiting modes [201]. (e) Densely [39] and (f) loosely [165] packed lattices. (g) A square lattice achieved with two square
sub-lattices bouncing out-of-phase [51]. (h) A more exotic Archimedean tiling [51] stabilized by the droplet in the center. (a) Reprinted bypermission from Springer Nature Customer Service Centre GmbH: Journal of Visualization. [161] © 2017. (c), (d) Reprinted figure withpermission from [201], Copyright (2019) by the American Physical Society. (e) Reprinted figure with permission from [39], Copyright(2007) by the American Physical Society. (f) Reprinted by permission from Springer Nature Customer Service Centre GmbH: Experiments
in Fluids. [165] © 2015. (g), (h) Reproduced from [51]. © IOP Publishing Ltd. All rights reserved.

stroboscopic model of Couchman et al [50], to be described in
section 5.2.
Drafting pairs or strings of walking droplets arise when asuccession of identical droplets walk in the direction of the line
joining them with a fixed distance between successive droplets.
These dynamic bound states were first examined experimentally by Filoux et al [43, 237], who reported that strings confined to a circular annulus may walk at speeds exceeding that ofindividual walkers at the same memory. The observed behavior of drafting pairs of walkers was captured by the simulationsof Durey and Milewski [33], who also examined their stability.
Eddi et al [41] demonstrated that when two bouncing dropsof unequal size are placed nearby, they interact through theircommon wave field in such a way as to self-propel, with theinter-drop distance fixed, and assuming one of a discrete set ofvalues. Owing to the asymmetry in the wave field along theirline of centers, the pair propagates through a ratcheting mechanism. Ratcheting pairs of bouncing droplets were examinedby Tang et al [58] using the liquid metal arrangement of Zhao
et al [198]. The authors drew the physical analogy with opticalratcheting, as arises when dielectric nanoparticles of unequal
size form optically bound states in a highly focused laser beam,leading to both self-propulsion and a discrete set of separation
distances [238, 239].
Galeano-Rios et al [47] revisited ratcheting droplet pairs
both experimentally and numerically. Their experiments
demonstrate that the quantized inter-drop distances of a ratcheting pair depend on the vibrational acceleration, and that as
the memory is increased progressively, the direction of theratcheting motion may reverse up to four times. Their simulations, based on the model of Milewski et al [44] (section 4.4),
highlighted the critical influence on the ratcheting motion ofboth the variations in the vertical bouncing dynamics and thetraveling wave fronts generated by each impact. The transient
wave generated by the partner drop creates a time-dependent
force that changes from repulsive to attractive as the transient
sweeps past. The reversal in the direction of motion of theratcheting pairs may arise due to either changes in bouncing
mode, as hypothesized by Eddi et al [41], or the subtle interplay between the traveling wave fronts and the droplet’s vertical dynamics. Valani et al [201] demonstrated that unequal
pairs of superwalkers may give rise to relatively high-speed
ratcheting (figure 5(c)) or a symbiosis in which a small drop
circles a large drop and so prevents the coalescence of its
partner (figure 5(d)).
5.2. Static bound states: identical pairs, rings and crystal
lattices

Crystalline structures may form from aggregates of identicalor nearly identical bouncers (figures 5(e)–(h)). Protière et al[37], Lieber et al [39] and Eddi et al [41] reported both stationary and spontaneously spinning droplet lattices. Eddi et al [51]achieved eight of the eleven Archimedean tilings of the planewith static bouncer arrays. Doing so required control of thedroplets’ relative bouncing phase: while some arrangements
were possible with all drops bouncing in phase (figures 5(e)and (f )), others required that neighboring drops be out ofphase, so could only be achieved with resonant (2, 1) bouncers
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Figure 6. Free and confined rings of bouncing droplets. Free rings exhibit a number of instabilities [240], including: (a) in-phase radial

oscillations, (b) orbital motion, (c) out-of-phase azimuthal oscillations and (d) out-of-phase radial oscillations. (e) 40 identical droplets
confined to a circular annulus [53]. The chain may destabilize to either (f) optical oscillations or (g) a propagating solitary wave. In (f) and(g), colors denote the droplets’ instantaneous angular velocity in rad s−1 and the inner ring radius is 24 mm. (a)–(d) Adapted withpermission from [240]. © Cambridge University Press. (e) Reprinted figure with permission from [53], Copyright (2020) by the American
Physical Society. (f), (g) Adapted figure with permission from [53], Copyright (2020) by the American Physical Society.

(figures 5(g) and (h)). Eddi et al [52] were the first to examine
the destabilization of crystal lattices in response to increasing
memory, and demonstrated that in both square and triangularlattices, instability sets in through a small oscillation of theindividual drops that may then propagate through the crystal
lattice like a phonon. As memory is increased progressively,
these oscillations grow until the lattice rearranges and eventually collapses into an irregular form. An accompanying theoretical model of a one-dimensional crystal lattice was used to
examine the onset of instability.
Couchman et al [50] examined the interaction of a pair ofidentical bouncing droplets, and characterized the dependenceof the system behavior on memory. At sufficiently low memory, the inter-pair spacing is arbitrary; however, at a criticalmemory, this distance becomes quantized. As the memory is
increased progressively, the interdrop distance changes weakly
until the static state gives way to a dynamic state characterizedby either colinear oscillations, promenading or orbital motion.
The presence of the partner drop can be either stabilizing ordestabilizing, depending on the droplet size. Larger drops went
unstable to collinear oscillations below the walking thresholdfor a single drop γ W . Smaller drops went unstable above γ W
to either orbital motion or side-by-side promenading motion,
depending on the initial interdrop distance.
Attempts to rationalize the behavior of identical pairs
prompted theoretical progress in the form of a variable-phasestroboscopic model [50]. The predicted pair stability changed
qualitatively through consideration of the variable phase:
constant-phase models always predict the onset of motion forγ < γ W . The authors used a quasi-static approximation of the

wave field of a stationary bouncer to derive a waveform witha purely spatial damping factor that is consistent with the farfield behavior of the wavefield defined in (11). They measured
directly the dependence of bouncing phase on forcing acceleration γ, drop radius R and local wave height hp . By measuring
the time-dependent vertical acceleration of a bouncing droplet,they inferred the contact force F(t) imparted from bath to drop,
and thus the phase parameters S and C defined in equation (10).
They then used Moláček and Bush’s [167] linear spring model(7), augmented by a term to account for the relic surface waves,in order to determine the dependence of S and C on γ, R and hp .
The resulting physical picture is that the bouncing phase varies
weakly with γ and hp so as to maintain a roughly constant
net upwards force, as required for periodic bouncing. Theirstudy highlights the fact that drops may interact through altering either the local gradient or amplitude of their neighbor’s
wave field. Through altering the wave amplitude at impact,
they change the impact phase, and so the evolution of both
drop and wave. Consideration of phase modulations is similarly critical in modeling the dynamics of free circular rings ofbouncing droplets, whose rich stability behavior has beencharacterized experimentally and theoretically by Couchman
and Bush [240]. In addition to the oscillations shown infigures 6(a)–(d), they reported rearrangement of the circular ring into regular polygonal forms, including squares andpentagons.
Barnes et al [241] examined theoretically the resonant
oscillations excited in a chain of bouncers when the drop at
one end is subjected to periodic forcing in the horizontal direction. They demonstrated that, at relatively high memory, the17Rep. Prog. Phys. 84 (2020) 017001

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Figure 7. The hydrodynamic spin lattice [54, 55]. (a) Plan view and schematic cross section of a ring of submerged circular wells, eachcontaining a single drop executing clockwise (blue) or anti-clockwise (red) motion on the surface of a vibrating fluid bath. A thin fluid layer
between wells enables wave-mediated interactions between neighboring droplets and the emergence of global order. Snapshots showing
collective (b) anti-ferromagnetic and (c) ferromagnetic order for different lattice spacings. (d) Rotating the system about a vertical axis withangular frequency Ω prompts a transition from antiferromagnetic to ferromagnetic states. Reproduced with permission from [55].

drops may oscillate with an amplitude larger than that prescribed, suggesting that the drops effectively extract energy
from the collective wave field. They also found that dynamic
stabilization of the chain could be achieved by sufficiently
high-frequency forcing.
Thomson et al [53] considered the instability of a ring ofbouncing droplets confined to a circular annulus (figure 6(e)).
Owing to the annular bounding geometry, the drops wereeffectively confined to a circle. For 20 drops, the onset of
instability was marked by small-amplitude optical oscillationsof the droplets, wherein adjacent pairs move azimuthally outof phase (figure 6(f )). For 40 drops, a solitary wave sweeps
through the array (figure 6(g)). At higher memory, counterpropagating waves arose; as additional wave modes appeared,
the ring packing collapsed and dislocations appeared in the lattice. The observed dependence of the form of instability on thedroplet number was rationalized theoretically [53, 242]. The
annular ring system exhibits several features of the Toda lattice [243], a canonical theoretical model of crystal vibrations[53], and so forges new links with solid-state physics.

space. Finally, Nachbin [234] noted that the wave-mediatedcoupling between the droplets is much richer than that positedin the canonical Kuramoto model for coupled oscillators [245].
Sáenz et al [54, 55] introduced an analog of an electronic
spin system, the hydrodynamic ‘spin lattice’, wherein walking
droplets are confined by an array of circular wells submerged
in a relatively shallow bath (figure 7). Each drop executes
circular motion within its own well, spinning either clockwise or counterclockwise with equal probability. When thevibrational forcing is sufficiently weak, each droplet’s wave
is confined to its own well. At higher memory, however, thewaves extend substantially beyond the well boundaries, resulting in wave-mediated interactions between neighboring drops.
To explore whether such wave-mediated spin–spin interactions can induce coherent collective order, Sáenz et al [55]considered a number of lattice geometries, and demonstrated
that three modes of analog ‘magnetization’ emerged according
to the lattice geometry and memory. Disordered paramagnetic
states arise when the spins of neighboring drops are uncorrelated. Antiferromagnetic or ferromagnetic order emerges whenneighboring droplets tend to have, respectively, opposite orequal spin.
In the ring geometry (figure 7), Sáenz et al [55] found an
optimal parameter regime in which the walker interactions
generate large-scale collective order. At sufficiently low memory, the walker wave fields were too weak for neighbors tocommunicate effectively, corresponding to a disordered paramagnetic state. As memory was increased, the influence of
the waves from the neighboring wells prompted the development of either antiferromagnetic (figure 7(b)) or ferromagnetic (figure 7(c)) order. Just as an antiferromagnetic material
may be transformed into a ferromagnetic state by imposing
a uniform magnetic field, a global transition from antiferromagnetic to ferromagnetic order was induced by applying system rotation (figure 7(d)). Consideration of the wave-mediatedinter-drop forces allowed the authors to describe the systemmathematically in terms of a generalized Kuramoto model for5.3. Coupled oscillators

Confining drops with bottom topography has allowed for the
study of wave-mediated interactions giving rise to synchronization and long-range order in walker lattices. Nachbin[234] used his two-dimensional, variable-topography model(section 4.7) to consider theoretically the one-dimensionalmotion of a pair of walkers trapped in separate cavities.
Between them, deep empty cavities facilitated wave-mediatedcommunication between the pair. Depending on the systemmemory, geometry and drop size, different correlations could
be induced between the pair. First, as observed in experiments [244], the drops could become perfectly synchronized,
moving horizontally back and forth in tandem, in- or out-ofphase. A second regime was reported in which the droplets
were not synchronized, but were statistically indistinguishable, with virtually identical signatures in position-momentum
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Figure 8. Orbital quantization [15] and chaotic pilot-wave dynamics [16] in a rotating frame. (a) Dependence of the orbital radius r0 on thebath’s rotation rate Ω at low memory (γ/γ F = 0.822), obtained both experimentally (dots, [16]) and theoretically (curve, [17]). The dottedcurve indicates the standard classical prediction for inertial orbits, r0 = u0 /2Ω, the offset from which has been rationalized with the boost
model [229] (section 4.6). (b) At higher memory (γ/γ F = 0.971), orbital quantization emerges. The orbits are color-coded according to theirstability, as assessed using the stroboscopic model [17] (section 4.3). Blue indicates stable orbits, green and red indicate orbits thatdestabilize via oscillatory and non-oscillatory instabilities, respectively. (c) In the high-memory limit (γ/γ F = 0.988), trajectories (inset)
become chaotic and are characterized by intermittent switching between weakly unstable orbits. Colormap: the probability distribution of
the radius of curvature R as a function of the rotation rate Ω. Bright segments indicate radii with high probability. Vertical dashed lines
correspond to the three probability distributions shown at right [16]. (a)–(c) Adapted with permission [16, 17]. © Cambridge University
Press.

coupled oscillators [245]. One thus expects that the hydrodynamic spin lattice will be a rich analog system capable of capturing many features of electronic spin systems and coupledoscillators, including spin waves [246].

However, as the memory is increased, the walker begins tointeract with its own wake, whose time-averaged form corresponds to circularly polarized Faraday waves centered at theorbital center (figure 8(b)). Fort et al [15] demonstrated bothexperimentally and numerically with the path-memory model(section 4.1) that the result is the emergence of quantized orbitsof radii Rn ∼ nλF /2, where n ∈ N. Owing to the analogous
form of the Coriolis force acting on a mass in a rotating frame
and the Lorentz force acting on a charge in a uniform magnetic
field, the authors drew the physical analogy between the quantized inertial orbits arising in the walker system and Landaulevels. Eddi et al [28] examined the influence of rotation on apair of orbiting walkers, showing that the inter-drop distance
was increased or decreased according to their sense of rotation relative to the ambient rotation, a hydrodynamic analog
of Zeeman splitting.
Walkers in a rotating frame were revisited experimentally
by Harris and Bush [16]. In addition to characterizing theonset of quantization at intermediate memory reported by Fortet al [15], they examined the onset of instability of the orbital
states, as well as the complex chaotic behavior emerging in thehigh-memory limit. Stable circular orbits were seen to give
way to wobbling orbits, then chaotic trajectories. The observed
stability of the circular orbits was rationalized theoretically by
Oza et al [17] via a linear stability analysis based on the stroboscopic model (section 4.3). A number of complex periodic

6. Orbital pilot-wave systems
Studies of the motion of walkers under the influence of externally imposed forces have allowed for an examination oforbital pilot-wave dynamics. Such studies have collectively
provided the first established paradigm for the emergenceof quantum-like statistical behavior from classical pilot-wave
dynamics, which is rooted in the chaotic switching between
weakly unstable quantized orbital states [5].
6.1. Analog Landau levels

Fort et al [15] were the first to consider the dynamics ofwalkers in a rotating frame (figure 8), a problem that has
subsequently been revisited both experimentally [16] andtheoretically [17, 219, 247]. When an object of mass m moves
at uniform speed u in a horizontal plane in a frame rotating uniformly with angular speed Ω about a vertical axis, it executes
an anticyclonic inertial orbit with radius u/2Ω, at which theradially outwards centripetal force is balanced by the inwards
Coriolis force. So it is for walkers at low memory (figure 8(a)).
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Figure 9. Double quantization of orbits and chaos in a central spring force F = −kx. (a) The mean radius R̄ and angular momentum L̄z of
periodic orbits captured over a range of γ/γ F and k [29]. Data points are color-coded
according to theorbits shown in the legend. (b), (c)Chaotic trajectories observed in experiments [34] at γ/γ F = 0.98 with (b) k/m = 4.44 s−1 and (c) k/m = 3.58 s−1 . (d) Time-evolutionof L̄z for the chaotic trajectory shown in (c). Colors correspond to sub-trajectories with (R̄/λF , L̄z /mu0 λF ) ≈ (1, 1) (yellow, clockwise
circular orbit), (1, −1) (green, counterclockwise circular orbit) and (2, 0) (pink, lemniscate). (e) A chaotic trajectory (gray) obtained using
the discrete-time model of Durey et al [33] contains the colored sub-trajectories shown. In (b), (c) and (e), scale bars denote λF . (a) Adaptedby permission from Springer Nature Customer Service Centre GmbH: Nature Communications. [29] © 2014.

L ∼ u0 m/k, where u0 is the walker’s free speed. At moderate memory and over a range of spring constants, a varietyof periodic and quasi-periodic states arose, including circles,
lemniscates, trefoils and papillons (figure 9(a)). These periodic
orbits were characterized in terms of their mean radius andangular momentum, and found to be roughly quantized in bothquantities. In the high-memory regime, the walker was seen toswitch intermittently between a number of these weakly unstable periodic orbits (figures 9(b)–(e)). At any given point inparameter space, typically two or three such states would beaccessible: in this sense, the chaotic motion could be considered as a superposition of periodic states. The physical pictureof a walker carving out then navigating its own slowly varying
topography was developed by Labousse et al [31], and applies
quite generally when the walker motion is confined.
The stability of circular orbits in a central force was treatedtheoretically by Labousse et al [220] using the constant-phasestroboscopic model (section 4.3). While qualitative agreement was obtained, the circular orbits were predicted to be
more unstable than those observed. This observed mismatch
between theory and experiment, also evident in the analysis

and aperiodic dynamical states arising at high memory were
captured by their simulations [219]. In the chaotic regime, thewalker motion is characterized by a chaotic switching between
weakly unstable circular orbits (figure 8(c)). The walker thus
has preferred radii of curvature; consequently, the histogramof the radius of curvature takes a peaked multimodal form
reminiscent of that arising in the analogous quantum system.
What would be considered a superposition of statistical statesin quantum mechanics may here be viewed, with the added
benefit of observable trajectories, as a superposition of weakly
unstable, quantized periodic orbital states.
6.2. Walkers in a simple harmonic potential

Perrard et al [29, 248] developed an experimental techniquefor encapsulating a drop of ferrofluid within a droplet, whichenabled them to impart external forces to walkers directly. By
applying a vertical magnetic field with a radial gradient, they
were able to examine walker motion in a central harmonic
potential. The system was characterized in terms of the system memory and the equivalent spring constant k of the linear
spring force, as determines the effective range of the walker,
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Rep. Prog. Phys. 84 (2020) 017001

Reviewof circular orbits in the rotating frame [17], is now thought tobe associated with the stabilizing influence of bouncing phasemodulations on the wobbling orbits [50]. Durey and Milewski[33] used their Faraday-wave model (16)–(20) to simulate thelong-time chaotic dynamics of a walker in a harmonic potential. They found that the walker’s chaotic dynamics resulted
from an intermittent switching between quasiperiodic trajectories consisting of ovals, lemniscates and trefoils (figure 9(e)),
as in the original experiments [29]. They subdivided the walker
trajectory into segments between successive radial maxima,
then computed the time-averaged mean radius R̄ and angular momentum L̄z of each sub-trajectory. Using the K-means
method to cluster the resulting data in the (R̄, L̄z ) plane, theyobtained a double quantization of R̄ and L̄z similar to thatreported in the experiments of Perrard et al [29]. A similar double quantization was evident in the simulations of Kurianski
et al [32], who noted that the double quantization arises
when the crossing time L/u0 is less than the memory time,and so relies on the walker interacting continuously with its
self-generated potential.
Durey et al [35] used the discrete-time model (section 4.4)
to simulate a walker confined to a line in the presence of alinear spring force. They showed that the walker’s dynamics become chaotic as the forcing acceleration is progressively increased, and that the route to chaos resembles the
Ruelle–Takens–Newhouse scenario. The authors also investigated the points where the drop reversed direction, and showedthat the statistical distribution of these turning points waspeaked at the minima of J 0 (kF x). Through a statistical analysis,
the authors showed that the walker’s long-time trajectory may
be interpreted as a random walk between these discrete turning
points. While a multimodal statistical signature reminiscent ofthat arising in the one-dimensional quantum harmonic oscillator emerges at intermediate time, the small variability in thestep length of this random walk leads to a smearing of thissignature in the long-time limit.

(section 4.3). The authors considered the dynamics of walking droplets acted upon by external forces, specificallyCoriolis, Coulomb, and linear spring forces. Initial conditions
for the drop and accompanying waveform were taken as thosearising for steady circular orbits. For each external force considered, an increase in γ destabilized circular orbits into wobbling and eventually chaotic orbits. The authors demonstrated
that the route to chaos following the destabilization of circular orbits depends on the form of the external force. For
the case of Coulomb and Coriolis forces, chaos sets in via
a classic period-doubling cascade [251]. In the case of acentral harmonic potential, the route to chaos is reminiscentof the Ruelle–Takens–Newhouse scenario, whereby the temporal power spectrum exhibits the successive appearance ofincommensurate frequencies with increasing memory, leading to the broadening of the spectrum characteristic of chaos[252–254].
Using numerical simulations of the stroboscopic model,
Budanur and Fleury [255] characterized the chaotic attractor of a walker moving in a central harmonic potential andshowed that it is composed of distinct regions corresponding todifferent unstable periodic orbits. They also showed that distinct chaotic attractors merge via a global bifurcation as thememory is progressively increased. Rahman and Blackmore[256] review the walking-droplet system from the perspective of the dynamical-systems-theory community. It is noteworthy that attempts to characterize transitions to chaos
experimentally face the difficulty that the transition typically
happens abruptly; for example, in the case of a walker in a
simple harmonic potential, over a span of Δγ/γ F ∼ 0.004[221]. Perrard and Labousse [34] drew a distinction between
the chaotic dynamics observed just at the onset of chaos, andthe intermittency observed in the high-memory limit. Specifically, they showed that the former exhibits characteristics of
low-dimensional chaos, as the dominant contributions to thewave field may be expressed using only a few modes. The latter
exhibits features reminiscent of noise-driven chaos: once thesystem departs a stable region of phase space, a rapid transition
drives it to another attractor.
Chaotic pilot-wave dynamics for drop motion in a rotating frame and a simple harmonic potential have key common
features. The form of fully chaotic walker motion appears
to be new for classical systems, and has provided the firstimportant paradigm for the emergence of quantum-like statistics in hydrodynamic quantum analogs [5]. At relatively low
memory, the drop executes one of several periodic orbits. At
high memory, these periodic states all destabilize, and thedrop switches between some number of them. One may thus
view the chaotic states observed in orbital pilot-wave dynamics as a superposition of weakly unstable, quantized periodic
orbits. While in quantum mechanics, systems are described interms of a superposition of statistical eigenstates, pilot-wavehydrodynamics suggests that these states are not incompatible
with the notion of particle trajectories in the form of unstable periodic orbits. Furthermore, the emergent statistics area reflection of the relative instability of these weakly unstable orbits. The system thus has features of the periodic orbit
theory proposed by Gutzwiller [257], who analyzed orbital

6.3. Chaos in orbital pilot-wave hydrodynamics

Chaos, specifically, sensitivity to initial conditions and theresulting lack of predictability, is evident in virtually all orbital
walker systems at high memory [5]. Chaos in hereditary systems is a subtle concept [249], and its possible significanceto quantum mechanics has been suggested previously [250].
While one usually characterizes chaos in terms of sensitivity
to initial conditions, in order to prescribe the evolution of ahereditary system, one must specify not only its initial conditions, but its history. In the walker system, one must prescribe
not only the initial conditions of the drop, but those of its spatially extended wave field, the form of which depends on thedrop’s history. While Tadrist et al [49] demonstrate that thelack of predictability in the walker system may be rooted in
uncertainty of the fast timescale responsible for wave generation, chaos would seem to be a more generic feature, as it also
appears in stroboscopic models.
Tambasco et al [221] characterized theoretically the onsetof chaos through the destabilization of circular orbits inorbital pilot-wave dynamics using the stroboscopic model21

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Figure 10. Dynamics and statistics of a drop in a circular corral. (a) In a corral of radius Rc = 10.1 mm = 2λF , double quantization ofangular momentum L̄z and mean orbital radius R̄ emerges in the intermediate memory regime (γ/γ F = 0.87–0.95) [14]. Gray circles
indicate sub-trajectories obtained by segmenting long trajectories between successive radial maxima [33], and periodic trajectories arecolor-coded according to the legend. The black crosses are the centroids found via K-means clustering. (b) In a corral of radius
Rc = 14.3 mm = 4.5λF , chaos ensues in the high-memory regime (γ/γ F = 0.989). Trajectories of increasing length are color-coded
according to droplet speed (in mm s−1 ) [10]. (c) The corresponding histogram of the walker’s position exhibits a multimodal statistical
pattern reminiscent of the quantum corral [259, 260], with oscillations on the Faraday wavelength λF . (a) Reprinted from [14], with thepermission of AIP Publishing. (b), (c) Reproduced figure with permission from [10], Copyright (2013) by the American Physical Society.

instabilities in the classical electrostatic n-body problem andused the computed classical trajectories to approximate energy
spectra. Similar approaches have been advanced in attempts
to characterize turbulence in terms of periodic subcomponents[258].

mean radius and angular momentum, reminiscent of walkers
in a simple harmonic potential [29] (figure 10(a)). They rationalized this similarity on the grounds that the boundaries act
effectively as a confining potential. They further examined theform of the mean wave field, noting that the symmetries ofthe periodic orbits were reflected in those of the associatedmean wave fields. The physical picture emerging from thesestudies of small corrals is similar to that arising in the orbital
pilot-wave systems discussed in section 6.
The most striking behavior in the hydrodynamic corralexperiments arose in relatively large corrals in the high memory limit, at memories within 0.2% of the Faraday threshold[10]. Here, the droplet follows a complex path marked byfrequent changes of direction prompted by interaction with
its pilot-wave field (figure 10(b)). Despite this erratic drop
motion, a robust statistical behavior emerged after approximately 30 min (figure 10(c)). The histogram of the drop position was comparable in form to the magnitude of the cavity’s
most unstable Faraday mode. The roots of this statistical signature were apparent in the speed map, which showed alternating concentric rings of anomalously high and low speed(figure 10(b)), which corresponded, respectively, to regionsof low and high probability in the droplet’s position histogram. The striking similarities between the quantum andhydrodynamic corral experiments are remarkable given thevast disparity of scales between the two (table 2).
The robust statistical forms emerging in the experimentsof Harris et al [10] motivated a number of theoretical studies of walker dynamics in bounded domains [11, 12, 207,247]. While certain features of the corral experiments havebeen captured, none of the theoretical models has satisfactorily captured its emergent dynamics and statistics. Gilet[11, 12] considered the discrete-time walker dynamics in a cavity with eigenmodes Ψk (x) satisfying a Neumann condition on
the boundary of the cavity. He considered the limit of zero drop

7. Analog quantum corrals
The quantum corral consists of a number of electrons (typically 60–80) moving on the surface of a metal, trapped by anarray of positively charged copper ions that serve as the fence
around the corral and discourage the electrons from escaping [259, 260]. Using a scanning tunneling microscope, thevoltage of the system may be measured, serving as a proxy
for the probability distribution function of the electrons. The
corral is appealing in that it represents a quantum system inwhich the wave function may thus be directly measured. The
observed statistical behavior is rationalized in terms of solutions of the time-independent Schrödinger equation with suitable boundary conditions. The observed wave forms can be
reproduced by superposition of several cavity modes whose
relative magnitudes are treated as fitting parameters [261].
7.1. Circular corrals

Harris et al [10] presented the results of an experimental investigation of a droplet walking in a circular corral, a deep liquid
region surrounded by a confining shallow layer. At the lowest
memories considered, the drop followed the outer boundary,executing circular trajectories. As the memory was increased
progressively, the circular orbits began to wobble, then gave
way to more complex periodic or aperiodic orbits such as stationary or drifting lemniscates or trefoils. This class of periodic
and weakly aperiodic orbits, prevalent in relatively small corrals, was classified in greater detail by Cristea-Platon et al[14], who noted that they exhibit a double quantization in
22

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Table 2. The characteristic scales of the quantum and hydrodynamic corral experiments indicate the remarkable range of the physicalanalogy.
Particle radius (m) Corral diameter (m) Particle frequency (s−1 ) Particle speed (m s−1 ) Statistical wavelength (m)Quantum
corral [259, 260]2.80 × 10−18

7.5 × 10−9

ω c = 1.60 × 1021

106

10 –9

Hydrodynamic
corral [10, 13]

3 × 10−4

2.86 × 10−250

2 × 10−2

4.75 × 10−3

inertia, and proposed the iterated map
xn+1 = xn − Kboundary condition on the corral’s edge. Building upon theirdiscrete-time model [33] (section 4.5), the authors derived a
map for the coupled evolution of the wave mode amplitudes,
drop position and velocity. They found that quantized circular orbits destabilize into precessing loops, rectilinear oscillations, and off-center lemniscates, comparable to those reported
experimentally in small corrals [10, 14], as the vibrational
acceleration is increased progressively. At high memory, thedrop switches intermittently between these weakly unstableorbital states. Although their model captured the reported formof the droplet dynamics at high-memory in relatively large corrals [10, 13], it did not capture the emergent wavelike patternin the walker histogram (figure 10(c)). The authors attribute
this shortcoming to their assumption that the walker’s vertical
dynamics is perfectly periodic, and conjecture that a model that
accounts for modulations in vertical phase might better capture
the walker’s observed statistical behavior.

ak,n ∇Ψk (xn ),kak,n+1 = μk Ψ∗k (xn ) + ak,n ,(23)

where 0 < μk < 1 plays the role of the memory parameter formode k, and K > 0 is a constant that prescribes the strengthof the wave–particle coupling. Gilet [11] first restricted his
attention to the 1D motion of a walker in a cavity with asingle mode Ψ(x). At low memory, the walker either settled
into a stable fixed point or executed a periodic orbit abouta fixed point. The periodic orbits destabilized as the memory was progressively increased, and the resulting chaotic trajectory could be decomposed into a sequence of intermittent
jumps between unstable states. Gilet found that the probability distribution of the walker position approximately satisfies∼ 1/|Ψ (x)|. Rahman and Blackmore [262] proved that the
fixed points in Gilet’s [11] model destabilized via supercritical or subcritical Neimark–Sacker (discrete-time Hopf ) bifurcations as the memory parameter is progressively increased.
They also proved that, as K is increased progressively, a periodic orbit may destabilize due to the appearance of a chaotic
strange attractor [263, 264].
Subsequently, Gilet [12] conducted numerical simulationsof equation (23) in a circular domain. Circular orbits stable at
low memory destabilized via a Neimark–Sacker bifurcation
as the memory was progressively increased. At high memory, there were no stable periodic attractors, and the walker
explored the entire cavity in a chaotic fashion. While theemergent statistics were qualitatively similar to those reported
by Harris et al [10], they differed quantitatively, presumably owing to the neglect of particle inertia, and differences
between the theoretical and experimental wave forms andboundary conditions. Gilet’s [12] simulations revealed thatchaotic walker trajectories were ballistic on short timescales
and diffusive on long timescales. The coherence distance, thelength scale beyond which the walker’s behavior was diffusive, was roughly λF /2. Gilet argued that the walker’s effective
diffusion coefficient then plays the role of /2m in quantum
mechanics, evoking Nelson’s Stochastic Dynamics [83, 84].
He further compared his model predictions for the momentum and kinetic energy to the analogous quantum observables,and suggested new links with experiments employing weakmeasurement in quantum systems [121].
Durey et al [207] simulated the pilot-wave dynamics of aresonant walker in a circular corral, using the wave modelin equations (13) and (14) augmented by a no-penetration

7.2. Elliptical corrals: mode superposition and statisticalprojection effects

Our understanding of the hydrodynamic corrals was advanced
considerably by the experimental study of Sáenz et al[13], who considered walker motion in an elliptical corral(figure 11). In the high memory limit, a robust statistical form
again emerged (figure 11(d)) from the correlation between
speed and position (figure 11(b)). However, two important new
inferences were made as to its form. First, while the instantaneous wave field is relatively complex (figure 11(a)), the mean
pilot-wave field (figure 11(c)) takes a simple form that is strikingly similar to the droplet histogram (figure 11(d)). Second,the emergent statistical form need not correspond to the most
unstable Faraday mode of the cavity, as was the case in the circular corral [10]. In the elliptical corral, the walker histogram
was expressible as a linear superposition of a small numberof modes: the presence of the drop could introduce additionalmodes that were the most unstable at nearby frequencies.
The study of Sáenz et al [13] was also important in demonstrating that pilot-wave hydrodynamics was viable in shallow water; thus, variations in bottom topography can be used
to influence the walkers. The concurrence of several statistical modes thus allowed for the possibility of using variations
in bottom topography to induce statistical projection effects.
Figure 11 indicates that the particle histogram (figure 11(d))
and the mean wave field (figure 11(c)) in an elliptical corralboth take a form that may be deduced by superposing two
distinct cavity modes, as is suggestive of a superposition of statistical states. Adding a well at an arbitrary position only introduced a localized perturbation to the histogram (figure 11(e));
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Figure 11. Walkers in an elliptical corral [13]. (a) The instantaneous wave field excited by the drop’s trajectory (dashed yellow curve).
(b) The chaotic trajectory is colored according to the instantaneous drop speed, revealing a correlation between position and speed. (c) Meanpilot-wave obtained by time-averaging the wave field over 30 min. The wave field is described by the superposition of two cavity modes,
specifically, the odd (1, 5) and even (4, 4) elliptical eigenfunctions (insets) of the Helmholtz equation with Dirichlet boundary conditions.
(d) Histogram of the walker position, deduced from the trajectory in (b). (e) Placing a submerged circular well at the midpoint of the uppersemi-minor axis leads to a localized disturbance. (f) When the well is placed at the left focus, amplification of the (4, 4) mode leads to theprojection of a high-density region toward the empty focus, an analog of the quantum mirage [265]. Reprinted by permission from Springer
Nature Customer Service Centre GmbH: Nature Physics. [13] © 2018.

J 0 (kF |x|). The prevalence of such circular orbits has been
noted in several experimental and theoretical investigationsof orbital pilot-wave dynamics [15–17, 26, 29, 31, 207, 219,
220], and would seem to suggest an energetic rationale fororbital quantization. The convolution result (24) was generalized to the case of variable bottom topography by Durey et al[207].
Another key insight from the study of Durey et al [35]was that, in the high-memory limit, the instantaneous wave
field converges to its time-averaged form. This convergence, as
demonstrated in their simulations of one-dimensional walker
motion in a simple harmonic potential, underscores the importance of the mean wave field in the walker dynamics. In particular, it indicates that the mean pilot-wave plays the role ofa self-induced nonlocal potential that the walker navigates,
reminiscent of the quantum potential in Bohmian mechanics(section 2.3). Efforts are currently being made to assess theextent to which such a physical picture describes the walker
dynamics in other settings. Notably, while the mean-pilotwave potential may be simply characterized in two dimensional corrals (e.g. figure 11(c)), the convergence of theinstantaneous pilot-wave field to the mean was not evident,
even at the highest memory, owing to the influence of viscous
wave damping.

however, when placed at one focus of the ellipse, it favored
one mode over the other (figure 11(f )), and so induced a signal at the other focus, a nonlocal statistical projection effectreminiscent of the quantum mirage [265].
7.3. The nonlocal mean-pilot-wave potential

The observed correlation between the histogram of a droplet
walking in an elliptical corral and the mean wave field [13] led
Durey et al [35] to an important theoretical result. They proved
that, for a walker executing perfectly periodic impacts in an
unbounded domain, the time-averaged pilot-wave field h̄(x)may be expressed as the convolution of the walker position
probability distribution ρ(x) and the wave field of a bouncer,
hB (x):
h̄(x) = ρ(x)∗hB (x).
(24)
This result holds for both periodic and chaotic walker motion.
For circular motion along an orbit of radius r0 , the walker
distribution is ρ(x) = δ(|x| − r0 )/2πr0 , so the mean wave
field is given by h̄(r) = ATTFM J0 (kF r0 )J0 (kF |x|) [29, 266]: it has
a Bessel form, and an amplitude prescribed by the orbital
radius. Notably, this result indicates that the mean wave field
vanishes identically when the walker orbits along zeros of24

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Figure 12. Walker tunneling. (a) The walker approaches a submerged barrier [9]. (b) Experimental trajectories (red) reported by Eddi et al[52]: the walker crossed the obstacle 14 times in 110 attempts. (c) Simulations of walker tunneling between two cavities when the walker isconfined to a line [8]. A space-time plot of the walker (red) and pilot wave (blue) captures a tunneling event. (d) Trajectories measured in theexperiments of Tadrist et al [9], displayed in position-velocity phase space. Reflected and transmitted trajectories are indicated in black andred, respectively. Points indicate the location of each impact. Green inset: incoming velocity fluctuations, with a characteristic error bar
shown. Blue inset: the separation point between reflected and transmitted trajectories. (a), (d) Reproduced figure with permission from [9],Copyright (2020) by the American Physical Society. (b) Reproduced figure with permission from [6], Copyright (2009) by the American
Physical Society. (c) Reproduced figure with permission from [8], Copyright (2017) by the American Physical Society.

8. Walker-boundary interactions

50% chance of tunneling, they determined the conditions fortunneling. They applied the technique of Tadrist et al [49] for
inferring the precise point and time of drop impact. They thus
deduced that that the spatial dependence of the bouncing phase
is not determinant in walker tunneling. Rather, their study
shows that the tunneling is determined by minute variationsin the drop’s horizontal momentum at a distance of approximately three barrier widths from the barrier: faster drops crossthe barrier, slower drops do not (figure 12(d)).

When a walker interacts with submerged topography, its pilotwave is altered, its motion disturbed. Recent studies of walkerboundary interactions have led to a number of intriguing
analogs of optical and quantum systems.
8.1. Tunneling

Eddi et al’s [6] experimental investigation of a walker impinging normal to a submerged barrier demonstrated that walkers exhibit features of quantum tunneling on a macroscopic
scale (figures 12(a) and (b)). The authors demonstrate that thereflection or transmission of a walker over a submerged barrier
is unpredictable; moreover, the crossing probability decreases
exponentially with increasing barrier width, as in the case ofquantum tunneling [267, 268]. Walker tunneling was revisited theoretically by Hubert et al [7], who used the Rayleighoscillator model of Labousse and Perrard [232] (section 4.6)
to capture the observed crossing statistics.
Nachbin et al [8] used their two-dimensional, variabletopography model (section 4.7) to simulate walker tunnelingacross a shallow region separating two relatively deep cavities(figure 12(c)). Simulation of the chaotic dynamics arising athigh memory showed that the probability of walker tunnelingacross a shallow region decreases exponentially with barrier
width, as in the original experiments of Eddi et al [6]. Moreover, for a fixed barrier width, the tunneling probability was a
non-monotonic function of the forcing acceleration, attaining
a maximum for a critical value of γ: the forcing accelerationneeded to be high enough for the local pilot-wave to permit
tunneling, but not so large as to generate large-amplitude
standing waves in the target cavity that prevent it.
Tadrist et al [9] revisited walker tunneling experimentally with a view to assessing whether resolution of the fast
bouncing dynamics would render the apparently unpredictablebehavior predictable. In a parameter regime where there was a

8.2. Reflection, refraction and optical analogs

A walker approaching a discrete step in bottom topography
at an angle of incidence θ i may either be reflected by thestep, or cross onto the shallow layer above the step and continue to walk. Pucci et al [203] examined walker reflection
from a submerged planar barrier. Their experiments showedthat droplets exhibit non-specular reflection. A small range ofreflection angles (60◦ < θr < 80◦ ) arises, and the precise valueof θr depends only weakly on the system parameters, including θi , the walker speed and size, the step size and the memory.
The observed behavior was captured theoretically with simulations based on the model of Faria [22] (section 4.7). The
non-specular reflection indicates that momentum is not conserved in the direction parallel to the boundary, that a wavemediated net tangential force is applied to the drop during its
interaction with the step, an effect rationalized in the theoretical developments of Turton [233].
Bragg diffraction was first observed in patterns of x-rays
scattered off crystalline solids. The Bragg condition is met for
certain wavelengths and incident angles, and is associated witha sharp decrease in the reflected radiation [269]. Vandewalle
et al [57] considered walker motion in a circular annular channel with periodically varying bottom topography consisting ofsubmerged walls aligned perpendicular to the channel. They
demonstrated that an analog Bragg condition exists: when thespacing of the walls is λF /2, the time-averaged droplet speed
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Figure 13. Scattering of walkers through single and double slits, as reported in the experiments of Pucci et al [23]. (a) A walker with impact
parameter yi passes through a single slit of width W = 14.7 mm = 3.1λF and is deflected by an angle α. (b) Trajectories in the deterministic
regime, γ/γ F = 0.985 ± 0.002. (c) Top panel: the dependence of the deflection angle α on the impact parameter yi for the trajectories inpanel (b). Bottom panel: the corresponding probability density function P(α) of the deflection angle. (d) Trajectories in the chaotic regime,
γ/γ F = 0.998 ± 0.002, where the deflection angle α is independent of the impact parameter yi , as evident in (e). (f) Trajectories of a walker
passing through one slit of a double-slit arrangement in the chaotic regime, γ/γ F = 0.998 ± 0.002. Note the marked difference from the
trajectories in (d), which indicates the influence of the second slit. Adapted with permission from [23]. © Cambridge University Press.

decreases sharply. Filoux et al [56] investigated the dynamics
of a walker confined to long rectangular corrals of different
widths d. They found that the walker’s longitudinal (v y ) andtransverse (v x ) velocities both increase with d, with the ratio
vy2 /vx2  being maximized for d/λF ≈ 1.5–2, as defines the
optimal wave guide.
The Talbot effect arises in optics when monochromatic
light is shone through a diffraction grating [270]. The
intensity pattern of light at the grating recurs at integer
multiples of the Talbot length, as is defined in terms of the
wavelength of incident light, λ, and the spacing of the grating,−1
[271]. Sungar et ald, through ZT = λ2 1 − 1 − (λ/d)2[59] demonstrated an analogous effect with the Faraday system excited just above threshold when there is a linear arrayof partially submerged pillars spaced at 2λF . The images ofthese pillars are projected by distances corresponding to integer multiples of the Faraday–Talbot length, which has precisely the form of the Talbot length, but with λF in place of λ.
They further demonstrated that the projected images could trap
bouncers and walkers, thus providing a hydrodynamic analog
of optical trapping with the Talbot effect. Sungar et al [60]showed that by spacing the pillars a distance 3λF /2 apart, they
could capture effects analogous to those of alternating gratingphase in optics. They also considered circular arrays of pillars,
which allowed for tuning of the magnification of the projected
images.
8.3. Diffraction from slits

The possibility of the walking droplet system as a hydrodynamic quantum analogue was launched by the seminal study of
Couder and Fort [2], who reported that walkers exhibit singleparticle diffraction and interference when passing throughapertures between submerged barriers. They sent single walkers toward a submerged barrier with openings on the scaleof the guiding wavelength (figure 13(a)). As the drop passed
through the slit, its trajectory was deviated owing to the distortion of its pilot wave, a form of macroscopic particle diffraction. The impact parameter yi , defined as the location of theincident trajectory of the walker relative to the centreline of
the slit, was varied so as to uniformly span the slit, in order tobest approximate an incident plane wave.
The principal inferences of Couder and Fort’s [2] study
were threefold. First, the deflection angle α was independent26

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Reviewof the impact parameter yi . Second, the emerging statistical pattern corresponded roughly to the amplitude of the
Fraunhofer diffraction pattern of a monochromatic plane wave
impinging on the slits. Third, in the double-slit arrangement,
the wave interacts with both slits; thus, the trajectory of thewalker is altered by the presence of the second slit, an indication of single-particle interference on the macroscale. Theirexperimental results in the single-slit geometry were roughly
reproduced by numerical simulations in which the walls weremodeled by periodically spaced secondary wave sources that
locally damp the walker wave field [117].
The results of Couder and Fort [2] were contested on statistical grounds by Bohr and co-workers [19, 20], who pointed
out that in the double-slit arrangement, the limited amount ofdata (75 independent trajectories) was insufficient to conclude
that an interference pattern emerged. Specifically, the fit witha Gaussian distribution was as good as that with the amplitudeof the inferred Fraunhofer diffraction pattern. The experimental results of Andersen et al [19] were also at odds with thoseof Couder and Fort [2], in that they observed a strong correlation between the diffraction angle and the impact parameter yi .
Moreover, no coherent diffraction pattern was apparent in theirexperiments. Subsequently, Rode et al [24] reported direct
measurements of the wave fields in the vicinity of the slits,
which they found were only weakly altered by the presence ofthe second slit.
Pucci et al [23] revisited the single and double-slit experiments of Couder and Fort [2] with a refined experimental
set-up that allowed for a systematic characterization of thedependence of the walker behavior on drop size and memory.
Notably, their study made clear the importance of shielding
the system from air currents, a precaution not taken in previous experiments [2, 19, 272]. As the drop size and memory
were not reported in the original experiments of Couder andFort [2], it was not possible to reexamine the parameter regime
originally considered. In the bulk of the experiments reported
by Pucci et al [23], there was a strong correlation between
deflection angle and impact parameter (figures 13(b) and (c)).
This correlation vanished only at the highest memory considered, γ/γ F = 0.998, in a small-drop parameter regime characterized by chaotic walking and unpredictable trajectories(figures 13(d) and (e)).
In all cases considered, the system behavior was dominatedby the tendency for a drop to follow a path with a fixed angle
relative to the plane of the slits, the same angle arising in nonspecular reflection from a submerged barrier [203]. The threedominant central peaks apparent in the diffraction patterns of
Couder and Fort [2] were thus evident in the majority of theparameter regimes considered by Pucci et al [23]; however,the dependence of the number of peaks on the slit width apparent in Fraunhofer diffraction was not evident. Nevertheless, inthe double-slit geometry, a key quantum feature contested by
Bohr and co-workers [19, 20, 24] was recovered: the droplet
is influenced by both slits by virtue of the spatial extent andpersistence of its guiding wave field (figure 13(f )). The experimental behavior was well captured by the model of Faria [22](section 4.7): discrepancy between simulation and experimentarose only at the very highest memories, where the droplet

bouncing becomes chaotic, so the walker dynamics cannot be
expected to be adequately described by a model based on theassumption of periodic bouncing. Walker diffraction by single and double slits was recently revisited by Ellegaard andLevinsen [25], who undertook the most comprehensive exploration of parameter space to date. Their experimental results
were largely consistent with those reported by Pucci et al [23],but also revealed new, relatively rich diffraction behavior in the
double-slit arrangement.
Diffraction is a feature of waves, both on macroscopic
and microscopic scales. Due to the association of a particle
with a wave in quantum mechanics, single-particle diffraction
has generally been thought to be exclusive to the quantum
realm. However, diffraction may also be a feature of classical objects, provided they have an associated wave component.
For example, Schiebel et al [273] have shown that snakes slithering through an array of posts exhibit a form of diffraction.
One should not find it surprising that the walker diffraction
pattern is different from that arising in single-electron diffraction. First, there is little reason to believe that the walker’s
pilot wave has a form similar to that of an electron. Second,
the geometries are markedly different: in electron diffraction,
the characteristic gap width is typically a thousand de Broglie
wavelengths, while in the walker system, the gap width is comparable to a single Faraday wavelength. Third, the influence ofthe hydrodynamic pilot wave is limited by viscous damping.
Nevertheless, the mechanisms for both single-particle
diffraction and interference in the hydrodynamic system areclear. Diffraction arises owing to the distortion of the drop’s
guiding wave by the slit arrangement. Interference by thesecond slit may arise through its acting to alter the global
wave field, specifically that beneath the walker. Both mechanisms rely explicitly on the hereditary nature of the walker
system and the spatial delocalization of the walker associated with its pilot-wave field; moreover, both are limited by
viscous wave damping. Recent developments in pilot-wavehydrodynamics suggest the manner in which one might findricher diffraction behavior in a generalized pilot-wave framework (see section 9). Notably, in the special case consideredin hydrodynamic quantum field theory (HQFT) [88], to bedetailed in section 9.3, the pilot-wave field in the vicinity ofthe particle takes the form of a plane wave with the de Broglie
wavelength [89].
8.4. Walker scattering from pillars and wells

Harris et al [172] examined the interaction of walking dropletswith a submerged circular pillar. At low memory, simple scattering events, characterized by a relatively rapid deflection
by some angle, are the norm. However, at high memory thedrop departs the pillar along a path corresponding to a logarithmic spiral (figures 14(a)–(d)). The form of the spiral isindependent of the droplet’s impact parameter: there is a universal spiral emerging for a given drop size, pillar size andmemory. The pitch angle increases with vibrational forcing:
the trajectory approached but never reached a circular orbit in
the high-memory limit (figure 14(e)). Because the drop speed
remains equal to the free walking speed along the spiral, and27

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Figure 14. Walker scattering [172]. (a)–(d) Surface topography measurements of a walker interacting with a submerged pillar of radius
5 mm (black circle) [204]. The white curve indicates the walker’s trajectory: a rectilinear approach path transitions to an outgoing
logarithmic spiral. The color bar indicates the wave amplitude in microns. (e) A droplet spirals about a pillar with radius 2.5 mm. As γ/γ F
increases from 0.97 to 0.99, the drop spends more time on a spiral path. Adapted from [172], with the permission of AIP Publishing.

9. Generalized pilot-wave frameworkthe drop accelerates slowly along the spiral, the system was
ideally suited to application of the boost model (equation (21),[229]). An effective force due to the presence of the pillar wasinferred, and found to be a lift force, Fp = mw Ω × ẋp , proportional to the boosted walker mass, mw and the cross productof the drop’s velocity ẋp and instantaneous angular velocity
about the post, Ω. When the equivalent form of the Coriolis and Lorentz forces is considered, this memory-induced lift
force is seen to be suggestive of self-induction: the drop moves
as would an electric charge in the magnetic field generated
by its own current. This system presents a clear macroscopic
example of pilot-wave-mediated forces giving rise to apparent
action at a distance.
Sáenz et al [18] considered a droplet walking in a shallow
bath, interacting with a deep circular well (figure 15). Here,the walker was drawn inwards along an Archimedean spiraltoward the well center. At high memory, the inferred wellinduced force once again takes the form of a non-local lift
force. After the drop crosses the well center, it moves radiallyoutwards, with speed oscillations arising over a lengthscalecorresponding to the Faraday wavelength. A superposition ofan ensemble of incident trajectories (figure 15(b)) reveals thatthese speed oscillations leave a circularly symmetric statistical signature with the Faraday wavelength (figure 15(c)). This
arrangement provides a hydrodynamic analog of Friedel oscillations, the statistical signature of electrons on a metallic surface scattering off an impurity [274, 275]. While walker speed
oscillations have been reported elsewhere in both experiments[10, 13, 168, 215] and simulations [215, 276], their role in
the emergent statistical behavior of walking droplets had notpreviously been noted and may be more prevalent than previously believed. Indeed, the origins of the statistical form
emerging in the large corrals [10, 13] may be rooted in suchspeed oscillations induced by boundary collisions [208], in
conjunction with the walker reflection laws [203]. Thesein-line speed oscillations represent a new paradigm for the
emergence of a statistical signature with the wavelength of thepilot wave, one that does not rely on the presence of chaos.

We have detailed the three key features responsible for emergent quantum-like behavior in the hydrodynamic pilot-wave
system. First, the pilot-wave must be generated by the particle; a free particle thus responds only to its own field. Second,
there must be a memory of the particle path recorded in thepilot-wave field, which renders the particle dynamics hereditary. Third, there must be resonance between the particle andwave field, which ensures that the pilot wave field be quasimonochromatic. Having established these essential features of
the hydrodynamic pilot-wave system, one may define a newclass of dynamical systems with these features, and dispensewith other nonessential details, including peculiarities of thehydrodynamic pilot-wave system, with a view to broadening
the range of classical pilot-wave dynamics and to capturing
new quantum analogs.
9.1. Parametric generalizations

The exploration of a simple parametric generalization of thehydrodynamic pilot-wave system was proposed by Bush [3],and has been profitably explored by a number of investigators[27, 36, 48, 266, 276]. The constant-phase stroboscopic model(12) discussed in section 4.3 may be non-dimensionalized toyield
κ0 (1 − Γ)ẍp + ẋp = −th̃(x, t) =−∞2∇h̃(xp , t),(1 − Γ)2

J0 (|x − xp (s)|)e−(t−s) ds,(25)


where κ0 = kF m3 gA/2D3TF is the dimensionless mass andΓ = (γ − γ W )/(γ F − γ W ) the dimensionless forcing acceleration, that assumes the value Γ = 0 at the walking thresholdγ = γ W and Γ = 1 at the Faraday threshold γ = γ F [3].
The dimensionless mass κ0 varies over the limited range
0.9–2.2 for the parameter regimes explored to date in the28

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Figure 15. Hydrodynamic analogue of Friedel oscillations [18]. (a) Oblique view of a droplet trajectory (yellow dots) passing over a
submerged circular well (dashed white circle). (b) The drop spirals inward toward the well along an Archimedean spiral, then exits the wellradially. Color coding the trajectories according to the drop speed, u/u0 , indicates speed oscillations in the outgoing phase. The right panel
shows an ensemble of 449 droplet trajectories in the high-memory limit, γ/γ F = 0.99. (c) The radial speed modulations give rise to a
Friedel-like signature in the histogram of the droplet position. From [18]. Adapted with permission from AAAS.

hydrodynamic pilot-wave system, but may be assigned any
value in this generalized framework.
Oza et al [17] and Labousse et al [26] explored theoreticallythe possibility of hydrodynamic spin states, wherein the walker
executes a circular orbit despite the absence of an external
force. While it was previously shown [17] that, without external confinement [26], spin states are unstable in the laboratory,
Oza et al [27] assessed the stability of spin state solutions toequation (25) for all values of κ0 and Γ. They showed that the
spin states of radius r0 ≈ λF /2 could be stabilized for sufficiently small values of the dimensionless mass, 0  κ0  0.2,
for a range of forcing accelerations.
Tambasco and Bush [266] examined a walker interactingwith a spatially oscillating radial potential of Bessel form using
numerical simulations of equation (25). By suitable choice ofκ0 and Γ, they were able to find a parameter regime in which
the walker switched intermittently between weakly unstable
circular orbits, and settled into a statistically steady state but
never a stable orbit. They also deduced that, in this setting,the timescale of statistical convergence was an order of magnitude greater than the memory time. Valani and Slim [48] used
a variant of equation (25) as the basis for numerical simulations of two identical in-phase walkers. In addition to the orbiting and promenading modes observed in walker experiments(section 5.1), they discovered a variety of exotic trajectories(figure 16).
In the Hong–Ou–Mandel (HOM) interference experiment[277], two photons arrive at a beam splitter, and two detectors far from the splitter record the arrival of the photons. As
the distance between the photon source and detectors is varied
continuously, the probability that a photon is detected by both
detectors exhibits a dip and eventually vanishes, which is theso-called HOM effect. Valani et al [36] conducted a numerical study of two walkers launched toward a common point ofintersection. For a parameter regime (κ0 , Γ) comparable to thatof the walkers, they deduced the probability that the pair forms
bound states (either orbiting, promenading or drafting). Suchbinding events correspond to a dip in the probability of the pair
continuing unperturbed, so give rise to a statistical signatureakin to the HOM effect.

Finally, equation (25) was adapted by Durey et al [276]to examine the prevalence of in-line oscillations for onedimensional motion in this generalized pilot-wave framework.
They delineated distinct parameter regimes characterized byunderdamped oscillations, of the form reported by Sáenz et al[18] in their study of analog Friedel oscillations, periodic speed
oscillations of the form reported by Wind-Willassen et al [168]and Bacot et al [215], and periodic and aperiodic jitteringmodes. In the jittering modes, the droplet bounces in place
until its pilot wave grows sufficiently to propel it into a nearby
wave trough. The process may result in either rectilinear periodic motion, or aperiodic motion characterized by irregularreversals in direction. The latter results in a random-walk-likedynamics and emergent quantum-like statistics reminiscent of
those reported for particle motion along a line in the presenceof a spring force [35]. Moreover, it leads to a characteristic
diffusivity D ∼ u0 λF , a result that evokes Nelson’s stochastic dynamics [83, 84] (section 2.5). A similar random-walklike behavior was observed theoretically in simulations of thepath-memory model (6) by Hubert et al [216], who foundthat, at high memory, a walker’s motion is ballistic over shorttimescales, sub-diffusive over intermediate timescales and diffusive over long timescales. Finally, in-line oscillations werealso prevalent in the simulations of Durey [278], who reduced
the equations for one-dimensional pilot-wave dynamics to aform similar to the Lorenz system [279].
9.2. Further generalizations

Further generalizations of the pilot-wave framework (25)will yield additional insights into the viability of pilot-wave
mechanics as a classical dynamical underpinning of quantum statistics. An obvious possibility is altering the kernelin the wave force, as prescribes both the wave form and its
spatio-temporal damping. One could also relax the assumptionof perfect particle-wave resonance, and so include a richer
dynamics of the form explored in the variable-phase stroboscopic model [50]. Another possibility is the inclusion of astochastic background field, which would in principle allow
for one to encompass pilot-wave dynamics and stochastic
mechanics in a single theoretical framework. Extensions to29

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Figure 16. Simulations indicate the rich behavior of identical in-phase particle pairs accessible within the generalized pilot-wave framework(section 9.1) [48]. In gray regions, the pair becomes unbound. Colored regions correspond to the bound states depicted in the legend.
Adapted from [48], with the permission of AIP Publishing.

three-dimensions, while not difficult conceptually, have thus
far met with limited success.
There is little reason to believe that walker-boundary interactions are closely related to those arising at a microscopic
scale; thus, various possibilities for boundary conditions inpilot-wave dynamics should also be explored. Dubertrand
et al [21] presented a model of single-slit diffraction [2,19, 20, 23, 24] in which walkers are scattered by a pair of
nearby submerged barriers. They time-averaged the trajectory
equation (8) over the bouncing period, and considered a wave
field h of the form

particle’s inertial frame at the time of emission, the result being
an interference pattern different from that in the walker system. The authors show that the wave field generated by thecircular motion of an inertial walker along an orbit of radius
r0 admits resonant wave modes when the orbital perimeter
2πr0 is an integer multiple of the Faraday wavelength λF ,a condition analogous to the Bohr–Sommerfeld quantization
rule [282].
9.3. Hydrodynamically-inspired quantum field theory

Several authors have attempted to formulate a form of quantum dynamics based on insights gained from the walker
system. Andersen et al [19] explored the behavior of a dynamical system in which a particle locally excites a waveform
satisfying Schrödinger’s equation, then moves in response
to gradients of the phase of that field. Orbital quantization
was shown to arise naturally, along with an analog to the
Bohr–Sommerfeld quantization rule; however, they concluded
that their model was not capable of giving rise to behavior analogous to that arising in the quantum double-slit experiment.
Borghesi [87] proposed an elastic pilot-wave model wherein apoint particle moves within a non-dissipative elastic substrate.
The coupled dynamics of the particle and elastic medium aregoverned by a modified Klein–Gordon equation. When theparticle speed is small relative to the elastic wave speed, it
is proportional to the gradient of the phase of the wave, aresult that evokes de Broglie’s guidance equation. Shinbrot[283] examined the Klein–Gordon equation with an oscillatory potential as a model for a quantum particle emitting andabsorbing pilot waves. He concluded that bound state solutions with half-integer spin exist provided the particle rest
mass oscillates in time, another result evocative of de Broglie’s
physical picture [130]. Most recently, Drezet et al [284] presented a mechanical analog of quantum bradyons and tachyonsn−1

Im[G(x, xk )]e−(n−k)TF /TM .

h(x, tn ) = −A

(26)

k=−∞

Here, G is the Green’s function for the Helmholtz equation(Δ + kF2 )G(x, x0 ) = δ(x − x0 ) with Neumann boundary conditions on the obstacles and outgoing radiation boundary conditions at infinity. Their numerical simulations show that,while the walker does indeed scatter when passing through theslit, the deflection angles are substantially smaller than thosereported in the walker system [2, 19, 20, 23, 24].
Devauchelle et al [280] developed a similar pilot-wave
model, but replaced the δ-function in the Helmholtz equation
by different functional forms to account for the particle’s
finite size. They neglected the particle inertia, and positedthat the particle’s velocity is proportional to the local surface wave gradient. Their combined analytical and numerical approach enabled them to demonstrate the existence of aself-propelling state resembling rectilinear walking; moreover,particle interaction with boundaries lead to in-line oscillations.
Broadening the scope of pilot-wave dynamics even further,
Fort and Couder [281] proposed a theoretical model for a socalled ‘inertial walker’ which consists of a self-propelling particle that generates waves that continue to propagate in the30

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Table 3. Different pilot-wave theories put into the general framework of equations (27) and (28). The pilot-wave phase S is defined throughthe equation u = |u|eiS/ , where u is φ (for de Broglie’s double-solution theory [80]) or ψ (for Andersen et al [19]).

Model
Stroboscopic (12)Oza et al [205]
Faraday waves (13)–(15)Trajectory equationuLf(x, t)mẍp + D ẋp = −F∇h(xp , t)h

−1/T MJ 0 (kF |x|)mẍp + D ẋp = −F(t)∇h(xp , t)

 
h
φ


 2
Δ|ψ(x ,t)|mẍp = −∇ − 2m |ψ(xpp,t)|ψ

i
Schrödinger equation 2mΔ

0ψ

Schrödinger equation− i A(t)δ(x)

Milewski et al [44]Bohmian mechanics [81, 82]
Schrödinger particle
Andersen et al [19]de Broglie’s double
solution [80]Hydrodynamic QFT [88]ẋp =∇S(xp ,t)m∇Sẋp = −c2 ∂S/∂tẋp = −α∇φ(xp , t)

 
φ
φt
 
φ
φt

2νΔ
σ
Δ − g(t)
ρ

D
2νΔ


Klein–Gordon equation






1
0

0
c Δ − ωc2
2

0
−PD (|x|, t)/ρ

Unspecified−p sin(2ω c t)δ a (x)Klein–Gordon equationof the wave phase, indicating that the wave–particle coupling
considered in HQFT is consistent with that proposed by de
Broglie [80] (table 3). In the relativistic limit, the de Broglie
wavelength approaches the Compton wavelength, suggesting
the possibility of quantization and structure on the Compton
scale. For example, spin states in HQFT might correspond tothe classical model of the electron [144]. HQFT thus promises
the possibility of accounting for the emergence of quantization and a statistical signature on the de Broglie wavelength
for non-relativistic mechanics, and structure on the Compton scale for relativistic mechanics. Further explorations of
two-dimensional HQFT, extensions to three-dimensions, andcoupling to the ensemble interpretation of quantum mechanics [103], will determine its viability as a trajectory-based
description of quantum systems.

through consideration of a vibrating particle propelling itself
along a frictionless string.
Dagan and Bush [88] have extended de Broglie’s doublesolution mathematical program through positing the form of
the dynamic interactions of waves and particles on the Compton scale, including the mechanism for particle-induced wave
generation. They model the particle as a localized periodic
disturbance in the Klein–Gordon field at twice the Compton
frequency (table 3). As in pilot-wave hydrodynamics, the particle is propelled by gradients in the amplitude of its pilot-wave
field. Resonance is achieved between the particle and its pilot
wave, leading to self-excited motion of the particle. Like thehydrodynamic bouncing state above the walking threshold, thestationary state gives way to a dynamic state characterized
by quasi-steady self-propulsion of the particle. Notably, themean speed rapidly converges to a value consistent with thede Broglie relation, p = γ L m0 v = kB . Moreover, the particle motion is characterized by relativistic in-line oscillations
with frequency ckB and wavelength λB , reminiscent of theZitterbewegung featured in early models of quantum dynamics[143, 145]. Their results suggest a new trajectory-based interpretation of the quantum dispersion relation, ω 2 = ωc2 + c2 kB2 :the first term corresponds to the energy of internal vibration
and wave generation, the second to the kinetic energy associated with in-line oscillations. The hydrodynamic pilot-wave
system has made clear that in-line oscillations provide a mechanism for a statistical signature with the oscillation wavelength[18]. One can thus readily imagine how statistical signatureswith the de Broglie wavelength might emerge on the quantum
scale.
Analysis of this HQFT [89] has revealed that there is a
critical wave–particle coupling constant beyond which selfpropulsion ensues, the analog of the walking threshold. In two
dimensions, when strobed at the Compton frequency, the pilotwave form in the vicinity of the particle takes the form of aplane wave with the de Broglie wavelength. Consequently, thelocal gradient of the wave amplitude is proportional to that

9.4. General particle-wave framework

The bulk of the pilot-wave models reviewed herein, bothhydrodynamic and quantum, have the form
Dt [xp ] = G u(xp , t), ut (xp , t), ∇u(xp , t), Δu(xp , t), . . . ,(27)
ut = Lu + f (x − xp , t),u = (h, φ, . . . ).

(28)

The particle trajectory is described by equation (27), the pilotwave evolution by (28). In the trajectory equation, the timederivative operator Dt acts on the particle position xp , giving
rise to time-derivative terms that must be balanced by somefunction G of the set of variables u characterizing the pilotwave field, evaluated at the particle position. For hydrodynamic pilot-wave theory, these variables u include the wave
height h and possibly other quantities, for example the velocity potential φ of the liquid bath in the formulation of Milewski
et al [44]. In the evolution equation (28) for u, the wave variables, f denotes the localized particle-induced forcing and L
is a linear differential operator. We note that Duhamel’s principle, which states that the solution to an inhomogeneous linear
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evolution equation such as equation (28) may be expressed asa time-integral over the history of the forcing f, necessarily
implies that the memory of the particle’s trajectory xp (t) is
encoded in the wave variables u.
Table 3 shows how several of the hydrodynamic and quantum pilot-wave models may be put into this general pilot-wave
framework. The hydrodynamic pilot-wave theories describe
a balance between particle inertia, drag and the propulsive wave force, so Dt [xp ] = mẍp + D ẋp . In the stroboscopic
model (12), u = h, f (x, t) = J 0 (kF |x|) and L = −1/TM . In
the Faraday pilot-wave model, (13)–(15), u = (h, φ) andf(r, t) = (0, −PD (r, t)/ρ) is the localized particle-induced
forcing. The linear operator L contains the Dirichlet-toNeumann map for the velocity potential φ, D[φ|z=0 ] = φz |z=0 ,which reduces the partial differential equations (13)–(15) on
the domain z  0 to the free surface z = 0.
In Bohmian mechanics [81, 82], there is no drag: the particle inertia is balanced by the gradient of the quantum potential, whose form depends on the wave function ψ that evolves
according to Schrödinger’s equation. Notably, the Schrödinger
equation is not forced by the particle, so f = 0; that is, particles are driven by the wave function, but the origins of thiswave function are not specified [284]. This deficit was also afeature of de Broglie’s double solution theory [80], where thefunctional form of the particle-induced wave forcing f was
not specified. HQFT [88, 89] specifies this wave forcing in amanner informed by the hydrodynamic system.

bouncing plays the role of the particle vibration in de Broglie’s
theory, and the wave-droplet resonance that of the harmony ofphases in de Broglie’s mechanics [3]. The third is not explicit
in de Broglie’s mechanics, but is a feature of the hereditary
particle-wave interactions considered by Brillouin [288], whoshowed that for the case of a particle moving faster than its
wave field, orbital quantization may naturally emerge.
Three paradigms have now been established for the emergence of quantum-like statistics from pilot-wave dynamics ofthe form engendered in the walker system. The first relies on
the chaotic switching between weakly unstable quantized periodic dynamical states, the second on in-line oscillations of
the walker speed, the third on the droplet’s random-walk-likemotion across its pilot-wave field. The first two are evident in
the walker system, the third in theoretical models thereof. All
three rely on the highly structured, quasi-monochromatic formof the pilot-wave field, and so depend critically on the resonantinteraction between the droplet and its guiding wave.
The first paradigm arose from studies of walkers in closed
environments, their motion confined by either boundaries or
imposed forces. Examples include the small corral experiments [10, 14], where the walker range is prescribed by thecorral radius; walker motion in a simple harmonic potential[29, 30, 32, 33, 220], where the drop range
is prescribed by theeffective applied spring constant, L ∼ u0 m/k; and walker
motion in a rotating frame [15–17, 219], where the effective range is prescribed by the radius of the inertial orbit,
L ∼ u0 /2Ω. The quantized orbital states emerge in these systems owing to the dynamic constraint imposed on the droplet
by its pilot wave, when the crossing time, T C ∼ L/u0 , is lessthan the memory time, T M ∼ T d /(1 − γ/γ F ) [32]. This criterion implies that in crossing its domain, a walker is always
navigating a background potential of its own making. In thehigh-memory limit, the quantized periodic orbital states destabilize, and chaotic switching between these weakly unstablestates leads to the emergence of multimodal, quantum-likestatistics.
The second paradigm for the emergence of quantum-likestatistics was identified in the hydrodynamic quantum analog of Friedel oscillations [18]. A number of prior studies identified the walkers’ tendency to execute in-line speed
oscillations with a wavelength comparable to the Faraday
wavelength. Specifically, such oscillations were first reported
in the experiments of Wind-Willassen et al [168], and are also
apparent in the drop speed maps reported in the corral experiments [10, 13] (figures 10(b) and 11(b)). In-line speed oscillations were characterized experimentally and numerically by
Bacot et al [215], whose simulations suggest that they arise in a
relatively broad region of parameter space. The statistical signature associated with such oscillations is readily apparent in
the analog Friedel oscillations [18], but may also play a role inother settings, including wall reflection [203, 208], tunneling[9] and large corrals [10, 13]. The prevalence of in-line oscillations in the context of a generalized pilot-wave theory has
recently been examined by Durey et al [276]. Finally, in-line
oscillations with the de Broglie wavelength and the Compton
frequency, reminiscent of the Zitterbewegung motion of earlymodels of quantum dynamics [143, 145] are a robust feature

10. Discussion and future directions
The walking droplets represent a macroscopic example of aparticle moving in synchrony with its own wave field, one inwhich the particle-wave interaction is both visible and resolved
theoretically. Growing evidence suggests that its range as aquantum analog is limited by the viscous damping of the pilotwave field, and by the influence of drop inertia, features of
the walker system that may be sidestepped within a generalized pilot-wave framework. If microscopic particles are indeed
characterized by a high-frequency vibration at the Compton
frequency, the theoretical description of walking droplets may
yield insight into their dynamics. For example, it raises thepossibility that the runaway solutions in the Lorentz-AbrahamDirac equation may result from an unresolved dynamics on theCompton scale [233, 286, 287].
10.1. The emerging physical picture

Three features of pilot-wave hydrodynamics are critical tothe emergence of its quantum-like features. The first is theassociation of the particle with a self-propelling wave source.
The second is the resonance between the particle and wave,
which results in the highly structured wave field whose form
is responsible for the emergence of quantum-like features. The
third is path memory, the ability of the drop to store information on the bath surface, the feature responsible for the temporal non-locality of the drop dynamics that may result in themisinference of spatial nonlocality. The first two are explicitly features of de Broglie’s double solution theory: the droplet
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Reviewof HQFT [88, 89], suggesting the possibility of an analogous
dynamical feature on the microscopic scale.
The third such paradigm relies on the droplet interactingwith its wave field in such a way as to execute a random walkwith a step length comparable to λF , resulting in an effective diffusivity D ∼ UλF , where U is the mean walker speed.
This type of diffusive motion was first identified, in simulations of walkers in a corral, by Gilet [11, 12], who noted thesimilarity to the behavior of quantum particles in Nelson’s
stochastic mechanics [83]. Such motion has since been captured in high-memory simulations of free walkers [216] andone-dimensional walker motion in the presence of a confining spring force [35]. Such episodic motion has also beenreported in experimental investigations of walkers above the
Faraday threshold [157], and for subcritical two-frequency
vibrational forcing [200]. The emergent diffusive behavior
was also seen to accompany the chaotic ‘jittering’ mode captured theoretically by Durey et al [276] in their exploration of
the generalized pilot-wave framework, and similar dynamics
were reported by Dagan and Bush [88] in their simulations of
HQFT.
The hydrodynamic pilot-wave system is extraordinarilyrich, characterized by many disparate timescales ranging froma fraction of a second to hours, all of them accessible in thelaboratory. First, the Faraday period T F is the timescale ofwave generation, twice the bath’s driving period. Second, thememory time T M prescribes the longevity of the pilot wave.
Third, there is the translation timescale, T H = λF /u0 ; fourth,that of the drop’s horizontal acceleration, T A . For closed systems of extent L, there are three additional timescales: thecrossing time T C = L/u0 , the timescale of establishment of
the mean pilot-wave field, and the timescale of statisticalrelaxation, beyond which the system has converged to a statistically steady state. These last two both scale with thememory time, but the latter is typically an order of magnitude larger than the former [266]. One can also define arecurrence time T R , the time taken for a particle to cross its
own path. While T R /T C = 1 for one-dimensional motion, it
will increase dramatically in two and three dimensions. One
can well imagine that T R  T M is a requirement for quantum behavior, one that will be progressively more difficult
to satisfy in higher dimensions. Note that various theoreticalmodels have emerged in the limits where there is a disparity between these various timescales. For example, the stroboscopic model (12) [205] is based on the assumption thatT F  T H , the boost equation (21) [229] that T M  T A and theRayleigh oscillator model [232] that T M  (T A , T H ). Notably,
the great majority of these timescales are absent from the standard quantum mechanical description of microscopic systems,
where particle dynamics is not described. The explorationof the various asymptotic limits in the context of a generalized pilot-wave framework poses a substantial mathematicalchallenge.
The limitations of the hydrodynamic system are associatedwith two physical effects that are not features of de Broglie’s
pilot-wave theory, specifically, pilot-wave damping and particle inertia. The limiting influence of viscosity is immediately
clear from the fact that the quantum phenomena invariably

arise in the high-memory limit, when the pilot wave is most
persistent. The limitation imposed by finite droplet inertia issuggested by the stability of spin states [27] and the instability
to in-line oscillations [276] being enhanced in the small-inertia
limit. The inviscid, low inertia limit is currently being exploredin the walker-inspired HQFT [88, 89].
10.2. Reinterpretation

Hydrodynamic quantum analogs has provided a perspectivethat allows for a classical reinterpretation of at least someof the quantum language. The walker system represents aclassical realization of wave–particle duality as envisaged byde Broglie, wherein a real object has both wave and particle components. In the walker system, this duality is theroot of both the emergent quantized dynamical states andthe structured statistical forms. The superposition of statistical states reflects the coexistence of several weakly unstable periodic dynamical states [16, 29], a notion that is alsoprevalent in periodic orbit theory [257, 258]. Statistical projection effects may be induced by favoring one dynamical stateover another [13].
There is no intrinsic uncertainty in the walker system: it
is a classical system that evolves according to deterministic
laws. Nevertheless, in many settings, the dynamics is chaotic,
so predictability is lost. de Broglie’s interpretation of the quantum uncertainty relations is that they are an expression of anunresolved dynamics on the Compton scale [138]. Tadrist et al[49] proposed that the lack of predictability in the walker system may likewise be rooted in uncertainty in the fast timescaleof droplet bouncing. However, such indeterminacy is not a prerequisite for chaos, which is also apparent in simulations of thestroboscopic model [219, 221, 255], where vertical dynamics
are not explicitly treated. We note further that uncertainty inthe walker system can be introduced artificially through suitably intrusive measurement. Couder and Fort [2] pointed outthat the deflection of a walker passing through a slit could
be interpreted in terms of an effective position–momentum
uncertainty relation for the walkers: when a walker is confined spatially, the momentum in the direction of confinement
becomes uncertain due to walker deflection prompted by the
distortion of its pilot wave. The emergent physical picture isthus that uncertainty is not intrinsic to the walker system, but
may emerge as an expression of its chaotic dynamics, or in
response to intrusive measurement. In the absence of intrinsic
uncertainty in this coupled wave–particle system, the notionof complementarity may be seen as an unnecessary conceit.
The corral studies place in stark relief the non-problem ofwave-function collapse as it pertains to the walker system. If
the histogram presented in figure 10(c) were indeed a complete
description of the system, then the act of observation revealing
the droplet to be at a specific point would cause the spatiallyextended wave function to collapse instantaneously to a point.
Acknowledgment of an underlying dynamics obviates the needto fret over the superluminal nature of wave-function collapse.
Indeed, in this particular instance, the analog of quantum nonlocality might more sensibly be termed statistical nonlocality[289].
33

Rep. Prog. Phys. 84 (2020) 017001

Review

The walker system has also provided a number of examples where wave-mediated forces give rise to apparent action
at a distance. Through their subtle influence on the pilot wave
field, topographic anomalies such as submerged circular pillars [172] and wells [18] were seen in section 8.4 to give rise
to nonlocal lift forces that caused the drops to follow spiralpaths. In the walker double-slit experiment [2], the presence ofthe second slit was seen to have an influence on droplets passing through the first [23]. Without awareness of the dynamicalsignificance of the pilot-wave field and the itinerant hereditary
droplet dynamics, such forces appear to be spatially nonlocal.
Richard Feynman insisted on the impossibility of understanding how an electron’s path can be altered by the presence ofa slit through which it does not pass [289]. The plausibility of
the answer proposed some 60 years earlier by de Broglie hasreceived support from the bouncing droplet system. The electron would interact with both slits if it were dressed by a wave
field that spans and is influenced by both slits.
Another means of misinferring a spatially nonlocal dynamics in the walker system was illustrated by Durey et al [35],who demonstrated the nonlocal form of the mean-pilot-wave
potential. Specifically, they showed that for 1D walker motionin the presence of a central spring force, in the high-memory
limit the instantaneous pilot-wave field converges to its mean,
whose form depends on the walker’s statistical distribution.
Thus, in this instance, local wave-mediated forces engenderan effect on the walking droplets analogous to the quantum
potential in Bohmian mechanics. However, because the instantaneous wave-field differs from the mean, the drop is also
subjected to an additional force that sidesteps the nonlocality of Bohmian mechanics while simultaneously precluding
the need to invoke a stochastic background forcing. This reasoning would seem to lend further credence to the view that
Bohmian mechanics describes a mean quantum dynamics,
an inference supported by the agreement between Bohmian
descriptions of slit diffraction [120] and recent experiments on
weak measurement [121].
Taken as a collective, studies of bouncer and walker lattices[38, 39, 50–55, 234, 240, 242] forge new connections between
the walker system and the field of crystal vibrations, connections to be clarified and deepened through analysis of different crystal arrangements. Studies of one-dimensional coupledwalker oscillators [234] and two-dimensional spin lattices [55]have revealed the possibility of wave-mediated forces giving
rise to long-range synchronization and collective order. The
precise relation between classical correlation, statistical indistinguishability [234] and entanglement in the walker system isthe subject of ongoing investigation. Particular attention will
be given to the notion that, just as nonlocality may be an
expression of hereditary pilot-wave dynamics, entanglement
may result from two particles having a common memory ofa wave-mediated interaction that synchronized their internal
clocks.

us only if we insist that the statistical description of quantum
particles is complete. The walker system suggests a physical picture that is compatible with the ensemble interpretations of quantum mechanics [102, 103]; however, it extends
beyond the realm of interpretation in suggesting a form for theunderlying dynamics of individual particles. Comparison of
the walker system with Bohmian mechanics and de Broglie’s
original mechanics indicates that its dynamical richness renders it closer to the latter than the former. However, pilot-wavehydrodynamics has buttressed de Broglie’s conception by providing a tangible example of a vibrating particle creating aquasi-monochromatic pilot-wave field, and being guided bythat wave in such a way as to generate quantum-like statistics.
The physical picture suggested by pilot-wave hydrodynamics,
as engendered in HQFT [88, 89], comes replete with a mechanism for particle-induced wave generation, and sidesteps theneed to invoke either a nonlocal potential or a stochastic
background field.
While one remains free to consider the statistical description of quantum mechanics to be sufficient, it is becoming
increasingly evident that this description is not incompatible
with the notion of microscopic particles following trajectories.
An argument against the development of a complex theory ofquantum dynamics of the form suggested by the walker system is that it is not worth the effort. Occam’s razor is often
invoked: Why complicate the simple statistical theory of quantum mechanics with a dynamical theory that will undoubtedly
be complicated and difficult to implement? One does well toremember that such an effort would be supported by an analogous philosophical Occam’s razor: the development of a rational quantum dynamics would allow us to dispense with thevery notion of quantum interpretation, and restore a rationalview of the microscopic world.
Acknowledgments
J W M B gratefully acknowledges the financial support of
the National Science Foundation (NSF) through grant CMMI1727565. A O acknowledges the support of the Simons Foundation through the Collaboration Grant for Mathematics No.
587006.
ORCID iDs
John W M Bush https://orcid.org/0000-0002-7936-7256
Anand U Oza https://orcid.org/0000-0002-9079-9172
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10.3. Philosophy

The walker system reminds us that much of the inscrutability of quantum mechanics and its paradoxes are forced upon34

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Imaging of Friedel oscillation patterns of two-dimensionallyJohn W M Bush is a Professor of Applied Mathematics in MIT’s Department of Mathematics. He received
his BSc and MSc degrees in physics from the University of Toronto, his PhD in geophysics from Harvard
University. Following postdoctoral research at DAMTP, University of Cambridge, he joined the MIT faculty MIT in 1998. His research concerns the discovery and exploration of physical phenomena arising ineither natural or laboratory settings, and their elucidation via the application of mathematics. His work typically has both experimental and theoretical components, and is rooted in MIT’s Applied Math Laboratory.
He is a fluid mechanician whose research began in geophysics, but then shifted to interfacial flows andtheir appearance in biology. For the past decade, his research has been focused on hydrodynamic quantum
analogs.

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Anand U Oza is an Assistant Professor in the Department of Mathematical Sciences at the New Jersey
Institute of Technology (NJIT). He received his bachelor’s degree from Princeton University, master’s
degree at DAMTP, University of Cambridge, and PhD in mathematics from MIT. He was an NSF
postdoctoral fellow at the Courant Institute, New York University, and then joined the NJIT faculty in2017. His research is concerned with physical applied mathematics, with a focus on fluid mechanics
and soft matter physics. His work uses a combination of mathematical modeling, asymptotic analysis,
and numerical simulation, and is typically motivated by laboratory experiments.

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