Introduction to focus issue on hydrodynamic quantum analogs
John W. M. Bush, Yves Couder, Tristan Gilet, Paul A. Milewski, and André Nachbin

Citation: Chaos 28, 096001 (2018); doi: 10.1063/1.5055383
View online: https://doi.org/10.1063/1.5055383
View Table of Contents: http://aip.scitation.org/toc/cha/28/9
Published by the American Institute of Physics

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CHAOS 28, 096001 (2018)

Introduction to focus issue on hydrodynamic quantum analogs
John W. M. Bush,1,a) Yves Couder,2 Tristan Gilet,3 Paul A. Milewski,4 and André Nachbin5
1

Department of Mathematics, MIT, Cambridge, Massachusetts 02139, USA
Matière et Sytèmes Complexes, CNRS UMR 7057, Université Paris Diderot, Sorbonne Paris Cité,
75013 Paris, France3
Microfluidics Lab, Department of Mechanical and Aerospace Engineering, University of Liege, Allée de laDécouverte 9, 4000 Liège, Belgium
4Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
5
National Institute for Pure and Applied Mathematics (IMPA), Est. D. Castorina 110, Rio de Janeiro, RJ
22460-320, Brazil
2(Received 7 September 2018; accepted 9 September 2018; published online 27 September 2018)
Hydrodynamic quantum analogs is a nascent field initiated in 2005 by the discovery of a hydrodynamic pilot-wave system [Y. Couder, S. Protière, E. Fort, and A. Boudaoud, Nature 437, 208(2005)]. The system consists of a millimetric droplet self-propeling along the surface of a vibrating
bath through a resonant interaction with its own wave field [J. W. M. Bush, Annu. Rev. Fluid Mech.
47, 269–292 (2015)]. There are three critical ingredients for the quantum like-behavior. The first is“path memory” [A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi, and Y. Couder, J. Fluid Mech.
675, 433–463 (2011)], which renders the system non-Markovian: the instantaneous wave force acting
on the droplet depends explicitly on its past. The second is the resonance condition between droplet
and wave that ensures a highly structured monochromatic pilot wave field that imposes an effective potential on the walking droplet, resulting in preferred, quantized states. The third ingredient ischaos, which in several systems is characterized by unpredictable switching between unstable periodic orbits. This focus issue is devoted to recent studies of and relating to pilot-wave hydrodynamics,
a field that attempts to answer the following simple but provocative question: Might deterministic
chaotic pilot-wave dynamics underlie quantum statistics? Published by AIP Publishing. https://doi.
org/10.1063/1.5055383

I. INTRODUCTION

In 2005, Yves Couder discovered that a millimetricdroplet may walk on the surface of a vibrating fluid baththrough a resonant interaction with its own wave field.1,4This hydrodynamic pilot wave system has since generated
considerable activity in the physics and applied mathematics communities2,5 and drawn the attention of philosophersof science.6 Not only is it a rich new dynamical system,
but it extends the range of classical systems to include
several features once thought to be exclusive to the microscopic, quantum realm. It is, moreover, strongly reminiscent of the double-solution pilot-wave theory proposed by
Louis de Broglie,7,8 according to which quantum particles
have an associated vibration or internal clock responsible
for wave generation and move in resonance with the resulting guiding or “pilot” wave. Studies of pilot-wave hydrodynamics seek to assess the potential and limitations of thiswalking droplet or “walker” system as a quantum analog,
while exploring its connections with extant realist models of quantum dynamics,2,5 including de Broglie’s doublesolution theory,7,8 Bohmian mechanics,9,10 and stochastic
electrodynamics.11
Pilot-wave hydrodynamics is an example of hereditarymechanics,12,13 a class of dynamical systems in which initial conditions (even if known precisely) are insufficient tospecify the system evolution; one must also have knowledgea)Electronic mail: bush@math.mit.edu

1054-1500/2018/28(9)/096001/7/$30.00

of the system’s past.75 In the walker system, this nonMarkovian feature arises by virtue of the system’s “pathmemory”:3 the wave force acting on the drop depends explicitly on the droplet’s past trajectory, along the entirety ofwhich the bouncing droplet generates waves. As the vibrating
bath approaches its Faraday threshold (the critical vibrational
acceleration above which waves form even in the absence ofthe droplet), the droplet’s wave field becomes more persistent, so the system memory increases. When coupled with theresonance between droplet and bath that results in a highly
structured, monochromatic wave field, the path memory may
lead to the emergence of stable quantized states. Further
increasing the memory may result in the emergence of chaos
characterized by unpredictable switching between unstable
periodic states and ultimately the emergence of multimodal,
quantum-like statistics.
With the walking droplet system, a number of hydrodynamic quantum analogs (HQA) have been discovered in
the laboratory and investigated both experimentally and theoretically. Examples include tunneling across barriers14–16
and refraction from single and double slits.17–19 A number of static and dynamic bound states of multiple droplets
have been discovered, including crystal lattices,20 orbiting
pairs,21,22 ratcheting pairs,23 and promenading pairs.24,25 In
the context of orbital pilot-wave dynamics, walker motion ina rotating frame has yielded analogs for quantized Larmor
levels,26,27 Zeeman splitting,28 and multimodal quantum-likestatistics emerging in the chaotic regime.27,29 The stability of

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hydrodynamic spin states, characterized by a droplet trappedin a circular orbit by its own wave field, has been considered
in a number of studies29,30 and is revisited in this issue.31
Walker motion in a simple harmonic well is characterized
by orbital trajectories that are quantized in both energy andmean angular momentum.32,33 Coherent quantum-like statistics also emerge in the hydrodynamic analog of the quantum
corral,34,35 where statistical projection effects and an analog of
the “quantum mirage” have been reported.36 The Faraday system has also been explored as an optical analog; for example,
a narrow channel has been explored as a walker waveguide,37
and walker trapping in the vicinity of a linear array of pillars isreminiscent of particle trapping with the Talbot effect.38 The
majority of these hydrodynamic and optical analogs will berevisited in this focus issue.
A hierarchy of theoretical models have been developed
to rationalize various walker behaviors.26,39–43 In this focus
issue, Turton et al.44 review these models, providing valuable perspective in detailing their successes and failures. A
table summarizes the different theoretical models developedby the HQA community. Stroboscopic models, deduced byaveraging the equations of motion over the droplet’s bouncing
period,39 prescribe the horizontal motion of walkers assuming perfect synchronization between the droplet’s bouncing
motion and the resulting wave form. While the stroboscopic
models have proven to be relatively successful in describing single-droplet behavior,29,33,40,42,45 they have limitations
in describing the dynamics and stability of orbiting pairs22
and promenading pairs.25 A current area of focus in the HQA
community is thus resolving the variability in the verticalbouncing dynamics,43 a feature explicitly neglected in the
stroboscopic models. The importance of variability in the
bouncing dynamics is highlighted in three separate studies inthis special issue.46–48 Finally, the recent theoretical modelsof Nachbin et al.16 and Faria49 allow for robust treatment of
walker-boundary interactions19,50 and will both be called uponin this focus issue.
The manner in which quantum-like behavior emerges in
the walker system is well studied in the context of “closed”
systems, which arise when the drop is confined spatiallythrough either imposed boundaries or applied forces. Examples include walker motion in corrals34,36 or in the presence ofa central spring force,32,33 both of which will be revisited inthis focus issue, and walker motion in a rotating frame.26,27,29
In such closed systems, the droplet navigates its monochromatic wave field, which imposes a dynamic constraint on its
motion. At relatively low memory, this constraint gives rise
to stable quantized orbits. As the memory is increased progressively, the quantized orbital states go unstable via oneof the classic routes to chaos.35,51–53 At high memory, thewalker switches between unstable periodic orbits, resulting in
multimodal, quantum-like statistics.27,32 While some of thisbehavior is captured by the stroboscopic models, discrepancies suggest the significance of variable vertical dynamics. In
this issue, Perrard and Labousse48 revisit walker motion in aharmonic potential and postulate that uncertainty in the vertical dynamics may be responsible for the chaotic switching inclosed systems. The routes to chaos in this rich new dynamicalsystem are evidently relatively subtle.

Chaos 28, 096001 (2018)

The walker behavior in open systems is no less rich,
but the links with quantum systems are not as clear. For
example, the extent to which quantum-like behavior arises inthe diffraction of walkers from single and double slits remains
a point of contention.17–19 While the experiments of Pucci
et al.19 indicate that the walker behavior is predictable overthe majority of parameter space considered, they did findunpredictable, evidently chaotic, behavior at the highest memories considered, the origins of which are not well understood.
While Nachbin et al.16 elucidated how unpredictability arises
in tunneling between cavities, the manner in which it arises
in tunneling across a barrier in an effectively open system14
is similarly unclear. We note that Perrard and Labousse’s48
proposal that noise associated with variability in the bouncing
dynamics may be a source of chaos is expected to be equallyvalid in open and closed walkers and so may have some bearing on this class of open systems. In this issue, Valani et al.54
demonstrate the rich behavior of walker-walker collisions as
described with a stroboscopic model. Tadrist et al.47 demonstrate the importance of vertical phase variations as the sourceof unpredictability and chaos in colliding walkers. Finally,
Galeano-Rios et al.46 demonstrate that the perturbation of thevertical motion through the influence of the transient wave
generated by their partner is critical. Evidently, variability inthe vertical dynamics may prove to be a critical ingredientfor chaos in both open and closed hydrodynamic pilot-wave
systems.
Bush2 proposed the extension of existing pilot-wave
models to a generalized pilot-wave framework capable ofachieving behavior inaccessible to the hydrodynamic pilotwave system, a theme revisited here in the review of Turton et al.55 While certain features of the walker systemare essential for the apparent quantum-like behavior, others are evidently not. The simplest possible generalization isan exploration of a parameter regime beyond that accessible
in the laboratory, an approach taken in three studies in thisfocus issue.31,56,57 Other possibilities include the consideration of alternative particle-wave couplings or exploration ofpilot-wave dynamics in three spatial dimensions. The results
reported here indicate the vast range of dynamical behaviors
accessible with this generalized pilot-wave framework.
In what follows, we briefly outline the contents of thisfocus issue and categorize the contributions into three central
themes. First, we summarize studies of closed pilot-wave systems, including walker motion in corrals, a circular annulus,
and a simple harmonic potential. Second, we summarize studies of open systems, including walker-walker interactions invarious guises, walkers interacting with background wave
fields, and walkers interacting with submerged topography.
Finally, we report explorations of a generalized pilot-wave
framework, of dynamical systems inspired by the walker system whose range of behavior extends beyond it. We closewith a brief discussion of current perspectives in the field as
informed by this special issue.
II. CLOSED SYSTEMSIn their paper, “Transition to chaos in wave memory
dynamics in an harmonic well: Deterministic and noise-driven

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behaviour,” Perrard and Labousse48 present the results ofan experimental and theoretical investigation of the chaotic
dynamics of a walker in a harmonic well. The authors show
chaotic trajectories in which the droplet oscillates between
strange attractors that can be associated with the eigenstatesof the system. Two distinct transition mechanisms are identified. The first mechanism is an intermittency dynamics
typical of low-dimensional chaos. As such transitions havebeen captured theoretically with stroboscopic models,42,45,53
one may surmise that variability in the vertical dynamics
is not an essential ingredient for them. Conversely, the second mechanism, noise-driven chaos, relies explicitly on theinfluence of noise in the vertical dynamics. Specifically, it
results from the fact that the global Lyapunov exponent of
the system, although strictly negative, tends asymptotically tozero when the memory parameter is increased. The walker
then becomes acutely sensitive to the discreteness of droplet
impacts, which induces chaos. Both mechanisms yield insight
into the selection rules for the transitions between eigenstates.
In their contribution, “Walking droplets in a circular corral: Quantization and chaos,” Cristea-Platon et al.58 report theresults of an experimental investigation of walkers confined tocircular cavities. Prior studies of corrals34,36,51 have focusedon the statistics emerging in the fully chaotic regime arisingat high memory in relatively large corrals. Here, attention is
given to smaller corrals, where periodic and quasi-periodic
orbits are prevalent. These orbits exhibit a double quantizationin average radius and angular momentum reminiscent of that
arising for walker motion in a simple harmonic potential.32,33
The study thus serves to unify the studies of walker motion
in closed systems. In such closed systems, when the memory time exceeds the crossing time of the domain, the walker
is always encountering a perturbed wave field, so responding to its environment as it navigates its own wave field.45
In serving to drastically decrease the dimensionality of thesystem to a few wave eigenmodes, confinement is responsible for the quantum-like statistics in this and other closed
systems.
Durey et al.59 present the results of a theoretical investigation of a walker subject to a central force and confined to one-dimensional motion. The discrete-time numericalmodel42 makes use of the spectral decomposition of the wave
solution and velocity potential that collapses the dynamics
onto fundamental matrices. The authors rationalize a numberof regimes, including periodic quantized oscillations, chaotic
motion, and the emergence of wave-like statistics. For an
unbounded geometry, the authors deduce a simple relationship between the time-averaged (mean) pilot-wave field andthe statistics of the droplets. Specifically, the mean wave field
is proven to be the convolution of the droplet’s histogram andthe wave field of a stationary bouncer. They further report thatin closed systems such as walker motion in a one-dimensional
simple-harmonic potential, the instantaneous pilot-wave field
converges to the mean, which thus plays the role of an additional applied potential. The walker dynamics is then cast interms of a Langevin-like equation wherein the wave force has
a component associated with the mean-pilot-wave potential aswell as a stochastic element associated with the local perturbation of the instantaneous wave field from its mean. The study

Chaos 28, 096001 (2018)

represents a substantial breakthrough in linking the dynamics
and statistics of walking droplets. Moreover, it makes exciting new connections to both Bohmian mechanics9,10 and de
Broglie’s double-solution pilot-wave theory.7,8
In “Standard map-like models for single and multiple
walkers in an annular cavity,” Rahman60 takes a discretedynamical-systems approach in describing single and multiple
walkers confined to a circular annulus. Inspired by the kickedrotator, a canonical problem in chaos and quantum chaos, theauthor develops a simple pilot-wave model in which each
impact imparts a discrete kick to the droplet. The dynamical system then takes the form of the standard map (the mostwell-studied one-dimensional map) for the angular position of
the droplet, with the kick strength being the control parameter. The bifurcations and the transition to chaos arising withincreasing kick strength are characterized. The model is thenextended to the case of multiple droplets. The models are
simulated and exhibit several features reported in the experimental studies of strings of walkers in an annular cavity.61
Using dynamical systems techniques and bifurcation theory,the single droplet model is analyzed to prove results suggestedby the numerical simulations.
In “Walking droplets correlated at a distance,” Nachbin62
reports the results of simulations of two droplets confined
to separate wells and divided by empty wells. The droplets
communicate through their common wave field, and theintervening empty wells serve as a resonant transmission
line. The droplets exhibit correlated dynamical features, evenwhen separated by a considerable distance. Specifically, theposition-velocity histograms of the two droplets display statistical coherence, and the droplets’ complex distributions inphase space are statistically indistinguishable. Removing one
drop drastically alters the phase space signature, which theauthor argues is reminiscent of quantum entanglement. The
author forges provocative new links between the pilot-wavehydrodynamic system considered and both the Kuramotomodel of coupled oscillators63 and entanglement in stochastic
electrodynamics.11
III. OPEN SYSTEMSThe paper “The interaction of a walking droplet and asubmerged pillar: From scattering to the logarithmic spiral”
by Harris et al.64 presents a striking puzzle. Their experiments
demonstrate that droplets aimed at a submerged pillar scatterat low memory but execute an expanding logarithmic spiralaround the pillar at higher memory. The form of the spiralis independent of the droplet’s impact parameter: there is auniversal spiral emerging for a given drop size, pillar size,
and memory. The system behavior is captured numericallyusing the theoretical model of Faria,49 in which the pillar
is treated as a region of decreased wave speed. Because thedroplet speed remains equal to its free walking speed along
the spiral, the system was ideally suited to application of theboost model,65 which was used to infer an effective force dueto the presence of the pillar. Remarkably, the pillar-induced
force is found to be a lift force proportional to the product ofthe drop’s walking speed and its instantaneous angular speed
around the post. This system thus presents a macroscopic

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example of pilot-wave-mediated forces giving rise to apparent“action at a distance.”
In 2008, Eddi et al.23 demonstrated that neighboring
droplets of unequal size interact through their common wave
field, forming a bound pair that self-propels through a ratcheting mechanism. In their contribution to this focus issue,
Galeano-Rios et al.46 present an integrated experimental andtheoretical investigation of such ratcheting pairs, detailing thedependence of the ratcheting behavior on the droplet sizes
and vibrational acceleration. Their experiments demonstratethat the ratcheters exhibit quantized inter-drop distances, withthis distance depending on the vibrational acceleration. For
a given pair, as the vibrational acceleration increases progressively, the direction of the ratcheting motion may reverse
up to four times. Numerical simulations based on recent
theoretical developments41,43 reproduce and rationalize thesystem behavior. Through demonstrating the critical importance of the vertical bouncing dynamics and specificallythe transient wave generated during impact on the ratcheting behavior, this study underscores the shortcomings of thestroboscopic models in capturing close-range droplet-droplet
interactions.
The paper “Bouncing droplet dynamics above the Faraday threshold” by Tambasco et al.66 is the first to systematically characterize experimentally the behavior of walkers
on a bath vibrated above the Faraday threshold. Several new
dynamical regimes are reported, including meandering, zigzagging, erratic bouncing, coalescence, and trapping. The
erratic bouncing state is similar to a random walk in which
the elementary steps are of the order of the Faraday wavelength. Similar random walks have been reported below the
Faraday threshold for walkers in confined geometries34,35 andopen systems.67,68 The study suggests the possibility of tuning the relative magnitudes of the background walker wave
fields, and that new HQAs may be discovered just above the
Faraday threshold, where the walker navigates a backgroundwave field prescribed by the system geometry. Such a scenario might be particularly useful in the context of diffraction,
as it would bear a strong resemblance to the physical pictures proposed by both Bohmian mechanics69 and stochastic
electrodynamics.70
In “Hong-Ou-Mandel-like two-droplet correlations,”
Valani et al.54 present the results of a numerical study of twodroplet correlations generated by a pair of walkers launchedtoward a common origin. By virtue of their pilot-wave coupling, the walkers may either scatter or form one of three
two-droplet bound states, specifically orbiting, promenading,or drafting pairs. The probability of such pairing is quantified as a function of the parameters of their pilot-wave
model. For certain parameters, the droplets may become correlated for certain initial paths and remain uncorrelated forothers, while in other cases, the droplets may never produce
bound states. The study suggests new directions in generating and rationalizing strongly correlated states in pilot-wavehydrodynamics.
Tadrist et al.47 present the results of a combined experimental and theoretical study of walker-walker interactions. In
their experiments, walkers are launched toward each other andeither scatter or lock into bound states, specifically orbiting

Chaos 28, 096001 (2018)

or promenading pairs. They introduce a new experimental
technique that allows them to deduce the walkers’ impact
phase from visualization of the wave field, a technique thatwill prove valuable in resolving the vertical dynamics inmany hydrodynamic pilot-wave systems. They demonstratethat the behavior during collisions cannot be predicted simply on the basis of the impact parameter; rather, the system ischaotic, the origins of the unpredictability being uncertainty
in the vertical bouncing dynamics. They introduce a theoretical model in which they model the droplet as an inelasticball riding a field of Faraday waves and demonstrate that thesystem behavior depends critically on the vertical dynamics
of the walkers, specifically, the relative phase of the walker
pair. Their study demonstrates that uncertainty in the verticalbouncing dynamics is responsible for the chaotic behavior in
their system. In so doing, it introduces a new means of promoting chaos through vertical phase variations in a number ofopen hydrodynamic pilot-wave systems, including single- anddouble-slit diffraction.17–19
IV. GENERALIZED PILOT-WAVE DYNAMICS

In their contribution to this focus issue, Oza et al.31
present theoretical results concerning the stability of hydrodynamic spin states. Under the action of a Coriolis force,
the walking droplet executes a circular orbit whose radius
becomes quantized at high memory owing to the droplet’sinteraction with its circularly polarized wave field.26,27 Oza
et al.71 rationalized the observed stability of these orbits andthe onset of chaos arising at high memory. They further
demonstrated theoretically that, if the walker’s wave force issufficiently strong, it may result in a circular orbit even in
the absence of rotation. While such hydrodynamic spin states
have not been reported in the laboratory,30 their stability is
considered here in the context of a generalized pilot-wave
framework2 based on the stroboscopic model of Oza et al.72
The authors characterize the parameter regime in which spinstates would be stable and demonstrate that the applicationof weak rotation would then result in an analog of Zeeman
splitting, as has been achieved experimentally with orbiting
pairs.28
The comprehensive theoretical and numerical study of
Valani and Slim,57 “Pilot-wave dynamics of two identical,
in-phase bouncing droplets,” explores the range of behaviorof this particular droplet pair configuration. Rather than constraining themselves to experimentally realizable parameters,
they explore the generalized pilot-wave framework proposedby Bush,2 wherein two control parameters prescribe the relative magnitudes of the inertial force, the drag force, andthe wave force. In so doing, they discover a remarkably richzoology of dynamical states, including experimentally achievable bound states such as in-line oscillations, drafting pairs,
orbiting pairs, and promenading pairs. More exotic boundstates hitherto unobserved in the laboratory were also discovered, including polygonal orbits of promenading pairs,
as well as wandering, lopsided, and reversing promenading
pairs. The study serves to demonstrate that pilot-wave systems
have a fascinatingly rich range of behaviors not necessarily
accessible to the hydrodynamic system.

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In their paper “Exploring orbital dynamics and trappingwith a generalized pilot wave framework,” Tambasco andBush56 examine pilot wave dynamics in the presence of acentral “well” of deeper fluid. Since the fluid in the well is
locally above the Faraday threshold, the well creates a Faraday wave field that acts as an external potential with a Bessel
form. Experiments show that this wave field may trap walking droplets on stable circular orbits. The authors then explore
theoretically a generalized pilot-wave framework2 in orderto obtain richer dynamical behavior. Specifically, they alter
the relative strengths of the drop inertia, the propulsive pilotwave force, and the imposed Bessel potential. Orbits may
then become unstable at high memory and transition to chaosthrough a path reminiscent of the Ruelle-Takens-Newhouse
scenario. A parameter regime is found in which the drop
switches chaotically between unstable circular orbits and ultimately converges to a statistically steady state. The form of
the emergent statistically stationary probability distribution of
the drops position reflects the relative instability of the circularorbits. The timescale of statistical convergence is characterized, and the mean wavefield in the chaotic regime is shown
to be related to the droplet histogram through the convolution
theorem of Durey et al.59 reported in this issue.
In their contribution, “Bouncing ball on a vibrating periodic surface,” Halev and Harris73 explore a dynamical systemrepresenting an elastic ball bouncing on a vertically vibrating
rigid surface with periodic topography. In addition to classicalperiod-doubling bifurcations, they report horizontal symmetry breaking that yields a stable “walking” regime. In thisregime, the ball motion consists of a horizontal translation of
one wavelength of the topography at each bounce. Particularattention is given to characterizing the influence of the bottom
topography, specifically the wave amplitude, wavelength, andwave shape, on the bouncing behavior.
A hydrodynamic analog to the optical Talbot effect wasrecently realized using a periodic array of pillars protruding from the surface of a vibrating fluid bath.38 When thepillar spacing is simply related to the Faraday wavelength,
repeated images of the pillars are projected in front of thearray. These self-images have their origins in the sloshinginter-pillar menisci that act as sources of Faraday waves. The
resulting self-images represent surface wave features that may
serve to trap walkers. In this issue, Sungar et al.74 build
upon this previous study in order to explore the emergenceof Faraday-Talbot patterns when the sloshing ridges between
successive pillar pairs are out of phase. A simple theoretical model rationalizes the observed self-image locations forboth linear and circular arrays of pillars. In the latter, arraycurvature allows for magnification and demagnification of theself-imaging pattern.
V. OUTLOOKThis focus issue reflects the current state of experimental, theoretical, and numerical investigations of hydrodynamic
pilot-wave dynamics. Experimental studies have introduced
new phenomenology46,48,64 that will motivate theoreticaldevelopments, as well as new experimental techniquesthat will find wide application in the HQA community.

Chaos 28, 096001 (2018)

Noteworthy is the technique of Tadrist et al.47 for resolving the drop’s impact phase through direct observation of
its wave field and the use of bottom topography to exceed
the Faraday threshold in a localized region and so induce aneffective background potential.56 The hierarchy of theoreticalmodels developed by the HQA community55 has been called
upon to explore both established46,48 and new60,62,64 walker
systems. The limitations of the stroboscopic models in a number of settings underscore the importance of resolving thefast dynamical scale,43 specifically that of droplet bouncing,in order to rationalize the dynamics and stability of boundstates,46 wave-induced trapping,38,66 and routes to chaos inboth open and closed systems.47,48
The contribution of Durey et al.59 represents a significant advance in our attempts to connect the dynamical andstatistical behavior of classical pilot-wave systems and tobetter understand their relation to realist models of quantum dynamics.8–11 While contributions to this special issue
demonstrate how pilot-wave-mediated forces can generate
both apparent “action at a distance”64 and the long-range statistical indistinguishability of droplet pairs,62 the definitionof entanglement measures in the walker system is a subjectof current interest and activity.6 This special issue has alsobroadened the range of the walker system beyond traditional
quantum31,48,54 and optical analog systems74 to touch uponquantum chaos,58,60 statistical mechanics,59 and the theory of
coupled oscillators.62 Finally, investigations of a generalized
pilot-wave framework31,56,57 provide early indication that thisvast theoretical landscape will be fertile soil for novel dynamical systems and new HQAs and so will serve to further expand
the phenomenological range of classical systems. It is hopedthat this focus issue will attract the attention and interest ofworkers from the dynamical systems community and inspiresome to join our exploration of this exciting new class ofproblems.
1

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3
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Mech. 675, 433–463 (2011).
4
S. Protière, A. Boudaoud, and Y. Couder, “Particle-wave association on a
fluid interface,” J. Fluid Mech. 554, 85–108 (2006).
5
J. W. M. Bush, “The new wave of pilot-wave theory,” Phys. Today 68(8),47 (2015).
6
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mechanics,” Found. Phys. 45, 458–472 (2015).
7
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10
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11
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M. Hubert, M. Labousse, and S. Perrard, “Self-propulsion and crossing statistics under random initial conditions,” Phys. Rev. E 95, 062607(2017).
16
A. Nachbin, P. A. Milewski, and J. W. M. Bush, “Tunneling witha hydrodynamic pilot-wave model,” Phys. Rev. Fluids 2, 034801(2017).
17
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macroscopic scale,” Phys. Rev. Lett. 97, 154101 (2006).
18
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Levinsen, and T. Bohr, “Double-slit experiment with single wave-driven
particles and its relation to quantum mechanics,” Phys. Rev. E 92, 1–14
(2015).
19
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interacting with single and double slits,” J. Fluid Mech. 835, 1136–1156(2018).
20
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in the bound states of wave interacting particles,” EPL 87, 56002(2009).
21
S. Protière, S. Bohn, and Y. Couder, “Exotic orbits of two interacting wave
sources,” Phys. Rev. E 78, 036204 (2008).
22
A. U. Oza, E. Siéfert, D. M. Harris, J. Moláček, and J. W. M. Bush, “Orbiting pairs of walking droplets: Dynamics and stability,” Phys. Rev. Fluids
2, 053601 (2017).
23
A. Eddi, D. Terwagne, E. Fort, and Y. Couder, “Wave propelled ratchetsand drifting rafts,” Europhys. Lett. 82, 44001 (2008).
24
C. Borghesi, J. Moukhtar, M. Labousse, A. Eddi, E. Fort, and Y. Couder,“Interaction of two walkers: Wave-mediated energy and force,” Phys.
Rev. E 90, 063017 (2014).
25
J. Arbelaiz, A. U. Oza, and J. W. M. Bush, “Promenading pairs of walking
droplets: Dynamics and stability,” Phys. Rev. Fluids 3, 013604 (2018).
26
E. Fort, A. Eddi, A. Boudaoud, J. Moukhtar, and Y. Couder, “Path-memory
induced quantization of classical orbits,” Proc. Natl. Acad. Sci. U.S.A. 107,
17515–17520 (2010).
27
D. M. Harris and J. W. M. Bush, “Droplets walking in a rotating frame:
From quantized orbits to multimodal statistics,” J. Fluid Mech. 739,
444–464 (2014).
28
A. Eddi, J. Moukhtar, S. Perrard, E. Fort, and Y. Couder, “Level splitting
at macroscopic scale,” Phys. Rev. Lett. 108, 264503 (2012).
29
A. U. Oza, D. M. Harris, R. R. Rosales, and J. W. M. Bush, “Pilot-wave
dynamics in a rotating frame: On the emergence of orbital quantization,”
J. Fluid Mech. 744, 404–429 (2014).
30
M. Labousse, S. Perrard, Y. Couder, and E. Fort, “Self-attraction into spinning eigenstates of a mobile wave source by its emission back-reaction,”
Phys. Rev. E 94, 042224 (2016).
31
A. U. Oza, R. R. Rosales, and J. W. M. Bush, “Hydrodynamic spin states,”
Chaos 28, 096106 (2018).
32
S. Perrard, M. Labousse, M. Miskin, E. Fort, and Y. Couder, “Selforganization into quantized eigenstates of a classical wave-driven particle,”Nat. Commun. 5, 3219 (2014).
33
M. Labousse, A. U. Oza, S. Perrard, and J. W. Bush, “Pilot-wave dynamics
in a harmonic potential: Quantization and stability of circular orbits,” Phys.
Rev. E 93, 033122 (2016).
34
D. M. Harris, J. Moukhtar, E. Fort, Y. Couder, and J. W. M. Bush, “Wavelike statistics from pilot-wave dynamics in a circular corral,” Phys. Rev. E
88, 011001(R) (2013).
35
T. Gilet, “Quantum-like statistics of deterministic wave-particle interactions in a circular cavity,” Phys. Rev. E 93, 042202 (2016).
36
P. J. Sáenz, T. Cristea-Platon, and J. W. M. Bush, “Statistical projection
effects in a hydrodynamic pilot-wave system,” Nat. Phys. 14, 315–319(2017).
37
B. Filoux, M. Hubert, P. Schlagheck, and N. Vandewalle, “Walking
droplets in linear channels,” Phys. Rev. Fluids 2, 013601 (2017).
38
N. Sungar, L. D. Tambasco, G. Pucci, P. J. Sáenz, and J. W. M. Bush,“Hydrodynamic analog of particle trapping with the Talbot effect,” Phys.
Rev. Fluids 2, 103602 (2017).
39
J. Moláček and J. W. M. Bush, “Drops walking on a vibrating bath:
Towards a hydrodynamic pilot-wave theory,” J. Fluid Mech. 727, 612–647(2013).
40
A. U. Oza, R. R. Rosales, and J. W. M. Bush, “A trajectory equation forwalking droplets: Hydrodynamic pilot-wave theory,” J. Fluid Mech. 737,552–570 (2013).
41
P. A. Milewski, C. A. Galeano-Rios, A. Nachbin, and J. Bush, “Faraday
pilot-wave dynamics: Modeling and computation,” J. Fluid Mech. 778, 361(2015).

Chaos 28, 096001 (2018)42

M. Durey and P. A. Milewski, “Faraday wave–droplet dynamics: Discretetime analysis,” J. Fluid Mech. 821, 296–329 (2017).
43
C. A. Galeano-Rios, P. A. Milewski, and J.-M. Vanden-Broeck, “Nonwetting impact of a sphere onto a bath and its application to bouncing
droplets,” J. Fluid Mech. 826, 97–127 (2017).
44
S. E. Turton, M. M. P. Couchman, and J. W. M. Bush, “A review of the theoretical modeling of walking droplets: Towards a generalized pilot-wave
framework” (submitted).
45
K. M. Kurianski, A. U. Oza, and J. W. M. Bush, “Simulations of pilot-wave
dynamics in a simple harmonic potential,” Phys. Rev. Fluids 2, 113602(2017).
46
C. Galeano-Rios, M. Couchman, P. Caldairou, and J. W. M. Bush, “Ratcheting droplet pairs,” Chaos 28, 096112 (2018).
47
L. Tadrist, N. Sampara, P. Schladheck, and T. Gilet, “Interaction of two
walkers: Perturbed vertical dynamics as a source of chaos,” Chaos 28,
096113 (2018).
48
S. Perrard and M. Labousse, “Transition to chaos in wave memory dynamics in an harmonic well: Deterministic and noise-driven behaviour,” Chaos
28, 096109 (2018).
49
L. M. Faria, “A model for Faraday pilot waves over variable topography,”
J. Fluid Mech. 811, 51–66 (2017).
50
G. Pucci, P. J. Sáenz, L. M. Faria, and J. W. Bush, “Non-specular reflectionof walking droplets,” J. Fluid Mech. 804, R3 (2016).
51
T. Gilet, “Dynamics and statistics of wave-particle interactions in a confined geometry,” Phys. Rev. E 90, 052917 (2014).
52
A. Rahman and D. Blackmore, “Neimark-Sacker bifurcations and evidence of chaos in a discrete dynamical model of walkers,” Chaos Solitons
Fractals 91, 339–349 (2016).
53
L. D. Tambasco, D. M. Harris, A. U. Oza, R. R. Rosales, and J. W. M. Bush,“The onset of chaos in orbital pilot-wave dynamics,” Chaos 26, 103107(2016).
54
R. N. Valani, A. C. Slim, and T. Simula, “Hong-Ou-Mandel-like twodroplet correlations,” Chaos 28, 096104 (2018).
55
S. E. Turton, M. P. Couchman, and J. W. M. Bush, “A review of the theoretical modeling of walking droplets: Towards a generalized pilot-wave
framework,” Chaos 28, 096111 (2018).
56
L. D. Tambasco and J. W. M. Bush, “Exploring orbital dynamics andtrapping with a generalized pilot-wave framework,” Chaos 28, 096115(2018).
57
R. N. Valani and A. C. Slim, “Pilot-wave dynamics of two identical, inphase bouncing droplets,” Chaos 28, 096114 (2018).
58
T. Cristea-Platon, P. Saénz, and J. W. M. Bush, “Walking droplets in acircular corral: Quantization and chaos,” Chaos 28, 096116(2018).
59
M. Durey, P. A. Milewski, and J. W. M. Bush, “Dynamics, emergent
statistics and the mean-pilot-wave potential of walking droplets,” Chaos
1, 10–20 (2018).
60
A. Rahman, “Standard map-like models for single and multiple walkers inan annular cavity,” Chaos 28, 096102 (2018).
61
B. Filoux, M. Hubert, and N. Vandewalle, “Strings of droplets propelled bycoherent waves,” Phys. Rev. E 92, 041004 (2015).
62
A. Nachbin, “Walking droplets correlated at a distance,” Chaos 28, 096110(2018).
63
J. Acebron, L. Bonilla, V. Perez, F. Ritort, and R. Spigler, “The Kuramotomodel: A simple paradigm for synchronization phenomena,” Rev. Mod.
Phys. 77, 137–185 (2005).
64
D. Harris, P.-T. Brun, L. Faria, and J. W. M. Bush, “The interaction of awalking droplet and a submerged pillar: From scattering to the logarithmic
spiral,” Chaos 28, 096105 (2018).
65
J. W. M. Bush, A. U. Oza, and J. Moláček, “The wave-induced added massof walking droplets,” J. Fluid Mech. 755, R7 (2014).
66
L. D. Tambasco, J. J. Pilgram, and J. W. M. Bush, “Bouncing
droplet dynamics above the Faraday threshold,” Chaos 28, 096107(2018).
67
N. Sampara and T. Gilet, “Two-frequency forcing of droplet rebounds on aliquid bath,” Phys. Rev. E 94, 053112 (2016).
68
M. Hubert, S. Perrard, M. Labousse, and N. Vandewalle, “Memory-driven
run and tumble deterministic dynamics” (unpublished).
69
C. Philippidis, C. Dewdney, and B. Hiley, “Quantum interference and the quantum potential,” Nuovo Cimento B 52, 15–28(1979).
70
J. Avendaño and L. de la Peña, “Matter diffraction through a double slit
by numerical simulation using a diffracted random electromagnetic field,”
Phys. E 42, 313 (2010).

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Bush et al.

A. U. Oza, D. M. Harris, R. R. Rosales, and J. W. M. Bush, “Pilot-wave
dynamics in a rotating frame: On the emergence of orbital quantization,”
J. Fluid Mech. 744, 404–429 (2014).
72
A. U. Oza, R. R. Rosales, and J. W. M. Bush, “A trajectory equation forwalking droplets: Hydrodynamic pilot-wave theory,” J. Fluid Mech. 737,552–570 (2013).
73
A. Halev and D. M. Harris, “Bouncing ball on a vibrating periodic surface,”
Chaos 28, 096103 (2018).

Chaos 28, 096001 (2018)
74

N. Sungar, J. Sharpe, J. Pilgram, J. Bernard, and L. D. Tambasco, “FaradayTalbot effect: Alternating phase and circular arrays,” Chaos 28, 096101(2018).
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E. Homer and A. R. Champneys (CRC Press, Boca Raton, FL), pp. 55–72.