Classical pilot-wave dynamics: The freeparticle
Cite as: Chaos 31, 033136 (2021); https://doi.org/10.1063/5.0039975
Submitted: 08 December 2020 . Accepted: 18 February 2021 . Published Online: 12 March 2021
Matthew Durey, andJohn W. M. Bush

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Classical pilot-wave dynamics: The freeparticle
Cite as: Chaos 31, 033136 (2021); doi: 10.1063/5.0039975
Submitted: 8 December 2020 · Accepted: 18 February 2021 ·Published Online: 12 March 2021
Matthew Dureya)

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CrossMarkand John W. M. Bushb)

AFFILIATIONS
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USAa)b)

Author to whom correspondence should be addressed: mdurey@mit.edu
bush@math.mit.edu

ABSTRACT
We present the results of a theoretical investigation into the dynamics of a vibrating particle propelled by its self-induced wave field. Inspiredby the hydrodynamic pilot-wave system discovered by Yves Couder and Emmanuel Fort, the idealized pilot-wave system considered here
consists of a particle guided by the slope of its quasi-monochromatic “pilot” wave, which encodes the history of the particle motion. We
characterize this idealized pilot-wave system in terms of two dimensionless groups that prescribe the relative importance of particle inertia,
drag and wave forcing. Prior work has delineated regimes in which self-propulsion of the free particle leads to steady or oscillatory rectilinear
motion; it has further revealed parameter regimes in which the particle executes a stable circular orbit, confined by its pilot wave. We herereport a number of new dynamical states in which the free particle executes self-induced wobbling and precessing orbital motion. We also
explore the statistics of the chaotic regime arising when the time scale of the wave decay is long relative to that of particle motion andcharacterize the diffusive and rotational nature of the resultant particle dynamics. We thus present a detailed characterization of free-particle
motion in this rich two-parameter family of dynamical systems.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0039975

The first macroscopic realization of a pilot-wave dynamics ofthe form proposed by de Broglie was discovered in 2005: an oil
droplet may “walk” across the surface of a vertically vibrating
fluid bath, propelled by its own wave field. This hydrodynamic
pilot-wave system exhibits many features previously thought tobe exclusive to the quantum realm, including orbital quantizationand emergent wavelike statistics. We here present a theoreticalinvestigation of an idealized pilot-wave system endowed with the
dynamical features of the walking-droplet system critical to its
quantum-like behavior. We pay particular attention to characterizing a new class of orbital states and the chaotic motion arisingwhen the pilot wave is long lived. We thus provide a comprehensive characterization of the free-particle dynamics arising in thispilot-wave system.

I. INTRODUCTION
A millimetric, bouncing droplet may self-propel on the surface of a vertically vibrating fluid bath, guided by its accompanying
Faraday wave field.1 The decay time of the subcritical Faraday waves

Chaos 31, 033136 (2021); doi: 10.1063/5.0039975
Published under license by AIP Publishing.

increases with the magnitude of the vibrational forcing, thus increasing the longevity and propulsive influence of the waves generated
along the droplet’s path. This decay time thus prescribes the extent
of the droplet’s “path memory.”2 In the long-path-memory limit, thequasi-monochromatic form of the guiding, or “pilot,” wave imposesa geometric constraint on the droplet dynamics that results in quantized dynamical states.3–6 The pilot wave also leads to the emergenceof quantum-like statistics, the origins of which may be characterized
in terms of three distinct paradigms:6,7 (i) chaotic switching between
weakly unstable quantized periodic dynamical states;4,8–10 (ii) in-line
speed oscillations over the length scale of the pilot wave;11 and (iii)
asymptotic diffusion arising from the droplet’s random-walk-likemotion across its pilot-wave field.7,12–14
While the walking-droplet system is the first classical realization of a pilot-wave dynamics of the form proposed by de Broglie,15,16
its limitations as a quantum analog prompted Bush5 to propose
the theoretical exploration of a new class of pilot-wave systems.
These idealized systems retain the salient features of the hydrodynamic system while side-stepping some of its complexity; moreover,they can be probed beyond the experimental regime of walking
droplets, yielding new insight into the potential for quantum-like

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Chaosbehavior in classical systems. Within the pilot-wave framework considered here,5 the dynamics of a free particle may be characterized interms of just two dimensionless parameters, allowing for a relatively
straightforward exploration of parameter space.
As the generalized pilot-wave system considered here is rooted
in the stroboscopic model of pilot-wave hydrodynamics developed by Oza et al.,17 it has certain features in common withthe walking-droplet system, including a steady rectilinear selfpropelling state1,2,7,17–21 and in-line speed oscillations.7,13,22,23 The
generalized pilot-wave system also has features that have been identified in various theoretical models of the hydrodynamic systembut have not been observed in the laboratory. Of particular interest to the present study are spin states,24–26 in which a particle
executes a circular orbit by virtue of the confining influence ofits pilot-wave field, and the erratic particle motion arising in thelong-path-memory limit.7,13,14,27–29
The generalized pilot-wave system also gives rise to dynamical
features that have been observed in neither the walking-droplet system nor theoretical models thereof. For example, when a particle isconfined by an oscillatory potential (of the form of a Bessel function), it may switch intermittently between different orbital states,
resulting in a dynamically unstable, statistically steady state. The
long-time dynamics thus give rise to a wavelike statistical form,30
consistent with the first paradigm for the emergence of quantumlike statistics in the walking-droplet system. When the interactionof particle pairs was considered in this generalized framework, avariety of periodic and chaotic bound states were shown to arise,31
demonstrating the richness of this new class of dynamical system.
Certain features of a free particle in the generalized pilot-wave
system evoke classical models of quantum systems. For example,
spin states are similar in form to the classical model of the electron.32
The in-line oscillations and jittering motion7 of the free particle
evoke the so-called Zitterbewegung, the trembling of a quantum
particle over the Compton time scale, a feature of early models ofquantum dynamics.33,34 Of particular interest here is the statisticalmanifestation of structured random walks arising in the long-pathmemory limit.7,13,14,27–29 Such random walks result in asymptotic particle diffusion, giving rise to a quantum-like statistical form28,29 anda physical picture reminiscent of Nelson’s stochastic mechanics,35 inwhich quantum dynamics is imagined as a diffusive process with diffusivity D = ~/(2m), where ~ is the reduced Planck constant and m
the particle mass.6
Hubert et al. 13 recently examined a new form of erratic pilotwave dynamics arising in their theoretical model of the walkingdroplet system. The droplet motion is characterized in terms of a
run-and-tumble dynamics,36,37 for which the resulting chaos is of the
Shil’nikov type, as arises when a homoclinic cycle interacts with asubcritical Hopf bifurcation.38,39 In the context of pilot-wave dynamics, the subcritical Hopf bifurcation corresponds to the instability ofthe steady rectilinear self-propelling state,7,13,14 the homoclinic cycle
to the transition of the particle between neighboring peaks of its pilot
wave.14 The authors further show that the erratic particle motion
exhibits asymptotic diffusion, with the corresponding diffusion
coefficient decreasing with increasing path memory. We will examine here similar run-and-tumble dynamics in the context of our generalized pilot-wave framework, characterizing the relative influenceof particle inertia, drag and wave forcing on the particle motion.

Chaos 31, 033136 (2021); doi: 10.1063/5.0039975
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We here provide a relatively comprehensive description of
the free-particle dynamics in our generalized pilot-wave system.
In Sec. II, we formulate the idealized pilot-wave system of interest and devise a new numerical method for efficiently evolving thepilot-wave system (Sec. II A). Using this numerical scheme, we identify and delineate regimes in which stable rectilinear and orbital
states arise (Sec. II B). In Sec. III, we characterize new wobbling andprecessing orbital states that arise just beyond the stability boundaryof the spin states. In Sec. IV, we explore the regime in which steady
self-propulsion is unstable, elucidating the rotational and diffusive
nature of the particle’s chaotic motion. Finally, in Sec. V, we discuss our results in the context of prior studies of the hydrodynamic
pilot-wave system and recent realist models of quantum dynamics.
II. CLASSICAL PILOT-WAVE DYNAMICS
We consider an idealized dynamical system in which a vibrating particle is propelled by the local slope of its guiding quasimonochromatic wave field and resisted by drag. Specifically, we
consider the evolution of a particle of mass m with position xp (t) attime t, where the particle vibrates with period T, generating axisymmetric standing waves about the particle’s instantaneous position.
Time-averaging the system over one vibration period19 yields a
trajectory equation of the formmẍp + Dẋp = −F∇h(xp , t),(1a)where D is a linear drag coefficient, F is the time-averaged force
exerted by the particle-wave interaction, h(x, t) is the time-averaged
pilot wave, and dots denote differentiation with respect to time. We
posit that the pilot wave, h(x, t), may be expressed as the superposition of quasi-monochromatic, axisymmetric waves generated along
the particle path. The waves decay exponentially in time over a timescale τ that prescribes the longevity of the particle’s path-memory.
We thus consider a pilot-wave formZA th(x, t) =J0 (k|x − xp (s)|)e−(t−s)/τ ds,
(1b)
T −∞
where A is the amplitude of the wave generated at each instant in
time, λ = 2π/k is the wavelength of the pilot wave, and J0 is theBessel function of the first kind with order zero.17
The stationary state, characterized by xp = constant andh(x) = (Aτ/T)J0 (k|x − xp |), destabilizes when the memory time, τ ,exceeds the critical threshold7,17
r
2DT
τ0 =
,
(2)
FAk2
beyond which sustained particle self-propulsion ensues. To characterize the free-particle dynamics arising beyond this critical threshold (τ > τ0 ), we non-dimensionalize the pilot-wave system (1) byscaling lengths with the inverse pilot-wavenumber, x ∼ k−1 , timewith the memory time at the onset of self-propulsion, t ∼ τ0 , and thepilot-wave amplitude by that at the instability threshold, h ∼ Aτ0 /T.
These rescalings result in the dimensionless pilot-wave systemκ0 ẍp + ẋp = −2∇h(xp , t),(3a)31, 033136-2

ChaosARTICLEh(x, t) =Z t−∞

J0 (|x − xp (s)|)e−µ(t−s) ds,(3b)

where κ0 = m/Dτ0 and µ = τ0 /τ are dimensionless parameters characterizing the particle inertia5,26 and wave damping,7,14
respectively.
As follows from (3b), the time scale over which the wave force,−2∇h(xp , t), is influenced by the particle path is 1/µ, and diverges
in the long-path-memory limit, µ → 0. To characterize the extent
of the path memory, we define 0 = 1 − µ, where 0 = 0 at theonset of particle self-propulsion and 0 = 1 is the limit at which
the wave damping vanishes. The parameter 0 may thus be seen tobe equivalent to the normalized vibrational forcing in the relatedhydrodynamic system, with 0 = 1 corresponding to the Faraday
threshold.5,26 We henceforth refer to the pilot-wave system (3) as
the generalized stroboscopic model. By varying the system parameters, we may explore a parametric generalization of Oza et al.’s17
stroboscopic model of pilot-wave hydrodynamics.5,26 As this modelis rooted in the walking-droplet system, we anticipate a number ofdynamical commonalities, which we summarize below. However, byexploring parameter regimes inaccessible in the walking-droplet system, we identify new stable orbital states arising in the generalized
stroboscopic model.
A. Numerical method
In order to explore the dynamics of the generalized stroboscopic model, we here develop a new numerical framework for
evolving the pilot-wave system (3). We first recall that the pilotwave system (3) exhibits translational invariance, a property that we
exploit by expressing the wave field in the frame of reference moving
with the particle.7,13 We then consider the basis decomposition13
h(x, t) = a0 (t)J0 (|x − xp (t)|) +

∞
X
n=1

2Re[an (t)8n (x − xp (t))], (4)where the complex basis functions 8n (x) = Jn (r)einθ are defined interms of the imaginary unit, i, and the polar coordinates (r, θ ), which
satisfy x = r(cos θ , sin θ ).
It remains to determine evolution equations for the complex
wave coefficients, an (t). As detailed in Appendix A, it follows from
Eq. (3b) that the wave coefficients evolve according toȧ0 − Re[żp a1 ] + µa0 = 1,
ȧn −

i
1h
żp an+1 − ż∗p an−1 + µan = 0,2(5a)∀n ≥ 1,(5b)where zp (t) = xp (t) + iyp (t) is the particle position, xp = (xp , yp ),represented in the complex plane and ∗ denotes complex conjugation. In the complex form, the particle evolution equation (3a)becomes
κ0 z̈p + żp + 2a∗1 = 0.

(5c)

We numerically evolve the complex representation of the pilot-wave
system (5) using a fourth-order Runge–Kutta method with appropriate truncation of the wave coefficients, an , details of which areprovided in Appendix A.

Chaos 31, 033136 (2021); doi: 10.1063/5.0039975
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B. Regime diagram
We proceed by characterizing the dynamics of the generalized
stroboscopic model in the entirety of the (κ0 , 0)-plane (see Fig. 1).
Beyond the onset of instability of the stationary state (0 > 0), theparticle executes rectilinear motion at a constant speed7,17
1/2
p
1 
(6)
u0 = √ 4 − µ2 − µ µ2 + 8
2in an arbitrary direction. For small 0, the response of the particle
when perturbed from steady rectilinear propulsion is characterized by overdamped oscillations.17 As 0 is increased, the perturbations become underdamped, exhibiting speed modulations over alength scale comparable to the wavelength of the pilot wave.7 For
0 > 0c , steady rectilinear propulsion destabilizes via a subcritical
Hopf bifurcation7 and erratic particle motion ensues.13 The particle exhibits growing speed oscillations until it reverses direction andis temporally trapped by its accompanying wave form,13 a dynamics
we further detail in Sec. IV.
Spin states24–26 may also arise in the generalized stroboscopic
model. Such states are characterized in terms of their orbital radius,
r0 , and angular frequency, ω0 , which satisfy the following pair ofalgebraic equations:10,24,26
Z ∞ 
ω0 s −µs
ω0 s 
sin
e ds,
(7a)
−κ0 r0 ω02 = 2
J1 2r0 sin
220
r0 ω0 = 2Z ∞0ω0 s 
ω0 s −µs
J1 2r0 sin
cos
e ds.
2
2

(7b)

As delineated by Oza et al.,26 spin states are stable only forκ0 < 0.183 (see Fig. 1). The corresponding orbital radius, r0 , iswell-approximated by the first zero of the Bessel function, J0 (kr0 )= 0, yielding r0 /λ ' 0.38, while the orbital speed, r0 ω0 , is generallycomparable to that of the steady rectilinear propulsion speed, u0 .26
Just beyond the boundary at which spin states destabilize, we
identify a small region of parameter space in which a new class ofdynamical states emerges, taking the form of stable wobbling andprecessing orbital trajectories (see Sec. III). We note that these relatively complex orbital motions all coexist with stable rectilinear
propulsion and may thus only be found with specific initializationof the pilot-wave system. Moreover, there is a minute portion ofparameter space (depicted by the white region in Fig. 1) in whichstable wobbling, precessing, and rectilinear propulsion all coexist,
highlighting the richness of this generalized stroboscopic pilot-wave
system.
As the pilot wave is central to the propulsion of the particle, it
is informative to compare its form for the different dynamical states.
As presented in Fig. 2(a), the pilot wave accompanying steady rectilinear propulsion has a horseshoe-like form ahead of the particle,
with an interference pattern apparent in the particle’s wake, reminiscent of that arising in the walking-droplet system.1,2,17,18,20,21,40 As
is evident in Fig. 2(b), the wave field accompanying orbital motion
is markedly different in form to that of rectilinear propulsion andserves to confine the particle to its circular trajectory.26 The evolution of the pilot wave accompanying wobbling, precessing, anderratic states is presented in the supplementary material.

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FIG. 1. Regime diagram delineating the emergent dynamical states, as classified in the legend. As 0 is increased, perturbations from the steady self-propelling statetransition from overdamped to underdamped oscillations. In the long-path-memory limit (0 > 0c ), the particle destabilizes into erratic motion. For κ0 < 0.183, stable spinstates may arise. Spin states may destabilize into stable wobbling and precessing trajectories, both of which exist within the white region. All such stable orbital states coexist
with stable rectilinear self-propelling states. White diamonds define the parameter values of the trajectories presented in the legend. The small speed perturbation, δv(t),presented in the legend is normalized such that δv(0) = 1.

FIG. 2. The form of the pilot wave, h(x, t), for two
coexisting stable states arising at 0 = 0.75 andκ0 = 0.15. (a) Steady rectilinear particle propulsion at speed u0 in the positive x-direction. (b) A
spin state, characterized by counterclockwise circular motion. In both panels, dashed lines denotethe particle path and dots denote the instantaneous particle position. The amplitude of the pilot
wave, h(x, t), is normalized by Aτ0 /T.

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III. WOBBLING AND PRECESSING ORBITS
We proceed by characterizing the form of the wobbling andprecessing orbital states arising near the boundary of the parameter regime containing stable spin states. We first observe that, as
evident in Fig. 1, wobbling and precessing trajectories only arisefor 0.79 < 0 < 0.91. While a large wave force is required to trap
the particle on an orbital path, these orbital states all destabilize
at sufficiently high 0. Moreover, wobbling and precessing statesarise only for κ0 < 0.2: as inertial effects are increased, the particle’s tendency for rectilinear propulsion is enhanced, resulting in thedestabilization of orbital motion.
In order to characterize wobbling and precessing orbital states,
we first translate the coordinate system so that the origin is the average position of the particle over its entire trajectory. We then define
the particle’s instantaneous radial displacement, rp (t) = |xp (t)|, andorbital period, P, which satisfy rp (t) = rp (t + P) for all t. Furthermore, we denote the moving average of the particle position over
one orbital period by the orbital center, xc (t), defined as9,10
Z
1 t+P
xc (t) =xp (s) ds.
P t
The relationship between the orbital center, xc (t), and the orbital
trajectory, xp (t), for wobbling and precessing states is illustrated inFig. 3. The orbital center of a wobbling orbit remains close to theorigin, tracing a trajectory similar in form to the particle path. In
contrast, the orbital center of a precessing trajectory resembles ahypocycloid, with characteristic cusps denoting the centers of thesuccessive loops of the particle trajectory.
We also characterize the orbital trajectory in terms of theinstantaneous radius of curvature, R(t), which necessarily decreases
in magnitude for tighter loops. The radius of curvature is defined asR=|ẋp |3,|ẋp ∧ ẍp |where ∧ denotes the vector product. Notably, for a circular orbit ofradius r0 , the orbital radius, radial displacement and radius of curvature are all equal (r0 = rp = R) and the orbital center is the origin(xc = 0).
As evident in Fig. 4, the radial displacement and radius of curvature of wobbling orbits (red curves) remain close to the orbital
radius, r0 , departing to a greater extent as κ0 is increased (with 0

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fixed). This behavior is suggestive of a supercritical Hopf bifurcationas κ0 is increased beyond the instability threshold of spin states. The
particle speed, v(t) = |ẋp (t)|, varies along the trajectory, achieving
a maximum value when the radial displacement, rp , is greatest (seeFig. 3). Moreover, the average orbital speed is typically comparable to the steady rectilinear propulsion speed, u0 , defined in Eq. (6).
For precessing orbits (blue curves in Fig. 4), the particle trajectory
exhibits relatively large variations in radial displacement, rp (t), andradius of curvature, R(t). Although the bounds of rp increase with
κ0 , we observe that the bounds of R remain approximately constant;indeed, the average value of R remains close to half the wavelength ofthe pilot wave. We observe that the speed variations along precessing trajectories are much greater than those along wobbling orbits;
however, as on wobbling orbits, the maximum particle speed arises
when the particle is at the extremities of its orbit (see Fig. 3).
We proceed by considering the angular displacement (indegrees), δθ , of the particle over one orbital period, P,δθ = arccos

xp (t) · xp (t + P).
r2p (t)(8)

We note that we have simplified the definition of δθ using the periodicity of rp (t). As presented in Fig. 4, the period, P, and angulardisplacement, δθ , both vary monotonically with increasing κ0 , theperiod increasing and the angular displacement decreasing. In general, the angular displacement is not a divisor of 360◦ , resulting in
the azimuthal drift of the orbital loops. We note the possibility ofan unstable branch connecting the wobbling and precessing orbital
states, as would provide a mechanism for the hysteresis observed as
κ0 is varied. Further investigation into this possibility will be left for
future consideration.
We define the instantaneous angular speed, ω(t), of the particle
as the product of its speed and the local curvature of the trajectory,
ω = v/R. We note that the angular speed is largest when the particle executes a sharp change in direction, vanishes for rectilinear
particle motion, and equals the orbital angular frequency, ω0 , forspin states. In Fig. 5, we present the evolution and bounds of thenormalized angular speed, ω/ω0 , for the orbital states. We observethat the instantaneous angular speed for wobbling states oscillates
about ω0 , resulting in a mean comparable to that of spin states. In
contrast, the angular speed for precessing trajectories exhibits far
greater variations and the resultant mean is appreciably less than

FIG. 3. Orbital trajectories arising at 0 = 0.85.
(a) Wobbling trajectory at κ0 = 0.13. (b) Precessing loops at κ0 = 0.18. In both panels, thetrajectory is color-coded by the particle speed normalized by that of a free particle, v/u0 . The orbital
center, xc (t), is superimposed (black curve), andthe arrow denotes the direction of particle motion.

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FIG. 4. Wobbling and precessing orbital states arising at 0 = 0.85, with κ0 in the range 0.104–0.197. (a) The average (dashed curves) and the upper and lower bounds (solid
curves) of the particle’s radial displacement, rp , for wobbling (red) and precessing (blue) trajectories. (b) The associated radius of curvature, R. (c) The radial displacementof the orbital center, rc = |xc |. (d) The normalized particle speed, v/u0 . (e) The orbital period, P. (f) The orbital angular displacement (in degrees), δθ, as defined in Eq. (8).

ω0 [see Fig. 5(c)]. The difference in the mean angular speed may be
attributed to the difference between the average radius of curvature
for wobbling and precessing trajectories, as quantified in Fig. 4(b).
However, for both wobbling and precessing trajectories, the maximum angular speed arises when the radial displacement of theparticle is greatest, coinciding with maxima in both particle speed
and curvature.
IV. ERRATIC DYNAMICS
We proceed by exploring the particle dynamics arising in theregime where steady rectilinear propulsion is unstable (0 > 0c ).

We characterize the dynamics in this regime in terms of two distinct phases, laminar and erratic13 [see Fig. 6(a)], as are related
to those arising in the canonical run-and-tumble dynamics.36,37
In the laminar (or “run”) phase, the particle motion is rectilinear but accompanied by growing speed oscillations [see Fig. 6(b)].
Eventually, the speed oscillations become so large that the particle reverses direction, triggering a switch from laminar to erratic
motion. The erratic (or “tumble”) phase is characterized by unpredictable changes in particle direction associated with the particle
navigating its complex wave field. When the particle eventuallyescapes the confining influence of its wave field, it resumes unsteadyrectilinear motion.

FIG. 5. The angular speed of wobbling and precessing orbital states arising at 0 = 0.85. (a) The instantaneous normalized angular speed, ω/ω0 , for a wobbling trajectoryarising at κ0 = 0.13. The arrow denotes the direction of particle motion. (b) The instantaneous normalized angular speed for a precessing trajectory arising at κ0 = 0.18.
(c) The mean (dashed curves) and bounds (solid curves) of the normalized angular speed for wobbling (red) and precessing (blue) trajectories arising for κ0 in the range
0.104–0.197.

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FIG. 6. Erratic pilot-wave dynamics arising at κ0 = 1, with 0 = 0.88 (red) and 0 = 0.94 (blue). (a) Sample particle trajectories. (b) and (c) Panels (i) and (ii) present
subtrajectories from (a), color-coded by (b) the instantaneous normalized speed, v/u0 , and (c) the instantaneous normalized angular speed, ω/ω0 . Arrows denote thedirection of particle motion. (d) Corresponding probability density function (PDF, dots) of the normalized angular speed, ω/ω0 , computed over a single simulation of duration
50 000τ0 [see Eq. (2)]. The mean value of ω/ω0 is represented by the dashed line for each value of 0. (e) Corresponding evolution of the mean-squared displacement,
1(t), averaged over N0 = 1000 similarly prepared systems. The long-time behavior, 1(t) ∼ 2Dt as t → ∞, is indicative of asymptotic diffusion, where D is the diffusion
coefficient.

We proceed by considering the influence of both thepath memory and particle inertia (characterized by 0 and κ0 ,respectively) on both the asymptotic diffusivity and rotational
nature [see Fig. 6(c)] of this chaotic dynamics in the context of thegeneralized stroboscopic model.
As the path memory increases, the more erratic nature of theparticle trajectory results in an overall increase in the average angular speed [see Fig. 6(d)]. In order to characterize the particle diffusivity, we consider a Monte-Carlo-like series of numerical simulationsof an ensemble of N0 similarly prepared systems. We then characterize the diffusivity in terms of the mean-squared displacement,
PN 0
2
1(t) = N−10
n=1 |xn (t) − xn (0)| , where xn (t) is the particle trajecth
tory for the n realization. The asymptotic diffusion coefficient, D,

Chaos 31, 033136 (2021); doi: 10.1063/5.0039975
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is defined so as to satisfy 1(t) ∼ 2Dt as t → ∞. In the long pathmemory limit, the prevalence of the erratic tumble phase decreases
the corresponding diffusion coefficient,13 as evident in Fig. 6(e). The
particle typically thus explores the plane less efficiently when thepath memory is increased.
For different values of the inertial coefficient, κ0 , we observe
similar trends in the diffusion coefficient and mean angular speed as
0 is increased (see Fig. 7). Specifically, the diffusion coefficient, D, ismaximal at the instability threshold of steady rectilinear propulsion(0 & 0c ) and decreases with increasing 0,13 while the mean angular speed is smallest at the instability threshold and increases withincreasing 0. We note that the largest values of the diffusion coefficient and the smallest values of the mean angular speed are achieved

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FIG. 7. Characterization of the erratic pilot-wave
dynamics arising for 0 > 0c (above the black
curve). (a) The normalized asymptotic diffusion
coefficient, D/λu0 , computed by averaging overN0 = 1000 similarly prepared systems. (b) The
normalized mean angular speed, ω/ω0 , averaged
over a single simulation of duration 50 000τ0 . In
both panels, the dots are simulation results andthe curves are visual guides, representative of contours. The squares correspond to the parametervalues shown in Fig. 6.

for similar values of κ0 . Specifically, these extrema are achieved
near the minimizer of the instability curve, 0c (κ0 ), corresponding
to κ0 ≈ 0.5.
V. DISCUSSION
We have explored the dynamics of the generalized stroboscopic
pilot-wave system,5,26 in which a vibrating particle is guided by its
self-generated wave field. A new numerical framework has allowed
for an efficient exploration thereof. Our investigation has furthered
the characterization of the free particle, delineating regimes in which
steady rectilinear propulsion and orbital motion are stable dynamical states. We have identified regimes in which new stable wobblingand precessing orbits arise. We have demonstrated that these stable orbital states coexist with stable rectilinear propulsion and may
only be accessed via specific initializations of the pilot-wave system.
In the precessing state, the particle executes periodic loops superimposed on a hypocycloid-like trajectory. If a spin state is taken to be aclassical analog of the electron,32 then these precessing orbits would
correspond to an orbiting electron. This analog electron would thenpossess two distinct forms of angular momentum that one mightrefer to as intrinsic and orbital.
We have also characterized the diffusive and rotational natureof the particle’s chaotic motion, as arises when 0 > 0c .13 In the
chaotic regime, the particle motion exhibits successive laminar anderratic phases. The particle motion becomes increasingly erratic as
the path memory is increased, resulting in an increase of the averageangular speed and a corresponding decrease of the particle’s asymptotic diffusion coefficient.13 These trends appear to be independentof the inertial coefficient, κ0 , which serves only to set the instability
threshold, 0c (κ0 ), of steady rectilinear self-propulsion.
As a caveat, we note that although our exploration of the(κ0 , 0) parameter space (see Fig. 1) is relatively comprehensive, wecannot rule out the possibility that more exotic dynamical statesmight be found from initializations of the pilot-wave system otherthan those considered here. Similarly, stable dynamical states mightarise within small pockets of parameter space when 0 > 0c , despite
the system’s propensity for chaotic particle motion at long pathmemory.
Our study has characterized new orbital states, specificallywobbling and precessing spin states, for the free particle. While similar orbital motions (spin, wobbling, and precessing states) also arise

Chaos 31, 033136 (2021); doi: 10.1063/5.0039975
Published under license by AIP Publishing.

in the hydrodynamic system,9,10 they are only stable when the droplet
is subjected to an applied force. It would thus be illuminating to
explore further the influence of applied forces on the particle motion
in our generalized pilot-wave framework. The aim of such investigations would be to extend existing numerical studies of the emergent
orbital and chaotic dynamics in the presence of Coriolis3,9,10,24 orcentral4,21,41,42 forces to parameter regimes beyond those accessible
with the hydrodynamic pilot-wave system. Of particular interest
would be the delineation of regimes in which multiple stable attractors coexist, as has already been shown to be the case for particle
motion confined to a line7,14,23,29 and for orbital dynamics.3,9,24 It
would also be intriguing to consider the symmetry-breaking effectof a Coriolis force on the particle’s rotational motion in the chaotic
regime and appeal to the analogy between the Coriolis and Lorentz
forces in order to characterize the emergent magnetization.
Finally, it is interesting to connect our work with recent pilotwave models of quantum dynamics inspired by the walking-droplet
system.43–46 In the hydrodynamic quantum field theory of Daganand Bush,43 a particle is propelled by the slope of a pilot wave
evolving according to the forced Klein–Gordon equation. As theform of this model of quantum dynamics is rooted in the walkingdroplet system, we anticipate a number of dynamical commonalities. Indeed, the simulations of Dagan and Bush of one-dimensional
particle motion revealed speed oscillations with the length scale ofthe pilot wave7,13,22,23 and an irregular jittering motion.7,14 An exploration of this hydrodynamic quantum field theory in two spatial
dimensions44 would allow one to determine whether analogous selfinduced orbital motion and stochastic dynamics might arise for freeparticles on the quantum scale.

SUPPLEMENTARY MATERIAL
See the supplementary material for movies of the pilot-wave
field and MATLAB implementation of the numerical scheme governing the evolution of the pilot-wave system.

ACKNOWLEDGMENTS
The authors gratefully acknowledge the NSF for financial support through Grant No. CMMI-1727565.

31, 033136-8

ChaosARTICLEAPPENDIX A: NUMERICAL SCHEME
To derive the complex set of differential equations governing
the evolution of the pilot-wave system (3), we first define the basis
functions 8n (x) = Jn (r)einθ , where (r, θ ) is the polar coordinate representation of x. For use in the following derivation, we note that
8n = (−1)n 8∗−n and
 
 
1
1
1
1
− 8n+1 (x).
(A1)∇8n (x) = 8n−1 (x)i−i22To express the wave field in the particle frame of reference, we first
write
Z t

80 (x − xp (t)) + (xp (t) − xp (s)) e−µ(t−s) ds.
h(x, t) =−∞Using Graf’s addition theorem,47
80 (x + y) =

∞
X

n=−∞

8n (x)8−n (y),we then express the wave field, h(x, t), as the basis decomposition13
h(x, t) =

∞
X

n=−∞

an (t)8n (x − xp (t)),

where the coefficients, an (t), are defined as
Z t
an (t) =8−n (xp (t) − xp (s))e−µ(t−s) ds.

(A2)

(A3)

−∞

−∞

where δm,n is the Kronecker delta. We evaluate the integral using
Eq. (A1), from which we obtain the system of ordinary differential
equations,
i
1h
(A4)
ȧn − żp an+1 − ż∗p an−1 + µan = δ−n,0 , ∀n ∈ Z.
2Here, zp (t) = xp (t) + iyp (t) is the representation of the particle position, xp = (xp , yp ), in the complex plane. We then exploit the reality
condition to restrict our attention to the coefficients with n ≥ 0: thebasis expansion (A2) for h(x, t) is recast to give Eq. (4), and the system and the system of equations [Eqs. (A4)] may be reduced to thesystem (5a) and (5b). Finally, we use Eqs. (4) and (A1) to transformthe particle evolution equation (3a) into a complex form, as given in
Eq. (5c).
To account for the stiffness associated with the κ0  1 regime,
we present a modification of the algebraic form of the particle trajectory equation (5c) in Subsection 1 of Appendix A. To initialize
the pilot-wave system for steady rectilinear propulsion or circularorbits, we determine specific algebraic expressions for the wave field
coefficients, an , as given in Subsections 2 and 3 of Appendix A,
respectively. The magnitude of the coefficients in each of these

Chaos 31, 033136 (2021); doi: 10.1063/5.0039975

states determines a suitable truncation of the basis functions. Specifically, for a truncation parameter δ > 0, we define the smallest
integer N such that |an | < δ for all n > N in a particular dynamical
configuration. We then set an = 0 for all n > N.
We note that the pilot-wave system (5) has no explicit dependence on the particle position, zp (t), reflecting the translational
invariance of the system. Furthermore, the quadratic terms in (5a)and (5b) arise due to our centering of the wave field in the particle
frame of reference and vanish when the particle is stationary. Whenthe particle motion is confined to a line (ẏp = 0), all the terms in(5) are real and we recover the set of evolution equations derived
by Durey et al. 7 We note that Perrard and Labousse48 exploited asimilar basis decomposition to evolve the pilot-wave system (3), butinstead expressed the wave field, h(x, t), in a fixed frame of reference. As a consequence of imposing a fixed origin on the system,
one cannot capture the translational invariance of the dynamics ina tractable manner. Finally, it is more efficient to evolve system (5)than recasting (3) as an integrodifferential equation for the particle
position, as we need not integrate over the particle’s prior trajectoryin order to compute the wave force.10 A more complete discussion of
the numerical efficiency of different modeling approaches (including continuous systems and iterative maps) may be found in Turton
et al. 49 and Bush and Oza.6
1. The overdamped limit

We note that the coefficients satisfy the reality conditionan = (−1)n a∗−n for all integers n.
To express the coefficients in differential form, we differentiate(A3) with respect to t, yielding
Z tȧn (t) = δ−n,0 − µan (t) + ẋp (t) ·∇8−n (xp (t) − xp (s))e−µ(t−s) ds,

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scitation.org/journal/cha

In the regime 0 < κ0  1, inertial effects arise on a fast timescale, resulting in a stiff dynamical system. To overcome the resultant stability issues arising when solving the pilot-wave system (5)numerically, we modify the algebraic form of the particle evolution
equation, introducing an integrating factor to evolve the particle
position analytically in the absence of a wave force.
We first define the time step δt and integration times tl = lδt
for all l ≥ 0. To evolve the system from time tl to time tl+1 , we follow
the methodology of Milewski and Tabak.50 Specifically, we multiply
the particle evolution equation (5c) by the integrating factor e(t−tl )/κ0 ,yielding2d  (t−tl )/κ0 żp e= − a∗1 e(t−tl )/κ0 .
dt
κ0We then introduce the modified velocity wp (t) = żp (t)e(t−tl )/κ0 . Consequently, the particle evolution in the time interval t ∈ [tl , tl+1 ] may
be expressed as the systemżp = wp e−(t−tl )/κ0 ,ẇp = −2 ∗ (t−tl )/κ0
a e
,
κ0 1

(A5a)
(A5b)

which couples with (5a) and (5b) to define the pilot-wave system.
Notably, the particle velocity satisfies żp (tl ) = wp (tl ) and żp (tl+1 )= wp (tl+1 )e−δt/κ0 . The pilot-wave system constituted by Eqs. (5a),(5b), and (A5) is then evolved numerically over the interval [tl , tl+1 ]using a fourth-order Runge–Kutta method.
2. Steady particle propulsion
We seek steady self-propelling states of the form zp (t) = z0
+ u0 teiφ , where u0 is the steady propulsion speed [see Eq. (6)] and31, 033136-9

ChaosARTICLEφ is the direction of propulsion in the complex plane, so xp (t)= x0 + u0 t(cos φ, sin φ). Seeking constant solutions to the wave field
coefficients [see Eqs. (5a) and (5b)] of the form an = Dn e−inφ gives
rise to the difference equations,−u0 D1 + µD0 = 0,u0
(Dn−1 − Dn+1 ) + µDn = 0,2∀n ≥ 1.

When coupled to the particle evolution equation (5c), this sysn
tem of difference equations has thep solution Dn = ρ2 , where22
2 −1/272
ρ = (µ + u0 )and 2 = (µ − µ + u0 )/u0 . We then truncate the wave field coefficients, an , based on their magnitude,|an | = ρ|2|n .
3. Orbital solutions
We proceed to seek orbital solutions to (5), specifically, zp (t)= z0 + r0 eiω0 t and an (t) = Cn e−inω0 t . From the integral representation of an (t) [see Eq. (A3)] with xp (t) = x0 + r0 (cos ω0 t, sin ω0 t), weapply Graf’s addition theorem47 to obtainZ ∞ 
ω0 s  inω0 s/2 −µs
Jn 2r0 sin
Cn = inee ds.
20
It is readily verified that an (t) satisfies the recurrence relations(5a) and (5b) for all n ≥ 0. It remains, therefore, to determine theorbital radius, r0 , and angular frequency, ω0 , using the equation of
motion for the particle. The substitution of the orbital ansatz into
the equation of particle motion [Eq. (5c)] determines that r0 and ω0satisfy the single complex equation,−κ0 r0 ω02 + ir0 ω0 + 2C∗1 = 0,or, equivalently,
iκ0 r0 ω02 + r0 ω0 = 2

Z ∞
0


ω0 s  −iω0 s/2 −µs
J1 2r0 sin
e
e ds.
2

By taking the real and imaginary parts of this equation, we recover
the orbital equation (7).
To justify truncating the wave field coefficients, an (t), whensimulating orbital states, we note thatZ ∞ Mnω0 s  −µs
e ds ≤,
Jn 2r0 sin|an (t)| = |Cn | ≤
2
µ
0
where
Mn = max |Jn (x)| ∼ √x∈[0,2r0 ]1
2π n er n0n

as n → ∞.

The approximation51 to the upper bound indicates that |an | exhibits
exponential-like decay for large n, which we exploit to truncate thewave field coefficients for orbital motion.
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data werecreated or analyzed in this study.

Chaos 31, 033136 (2021); doi: 10.1063/5.0039975
Published under license by AIP Publishing.

scitation.org/journal/cha

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