Quantumlike statistics of deterministic wave-particle interactions in a circular cavity

Abstract

A deterministic low-dimensional iterated map is proposed here to describe the interaction between a bouncing droplet and Faraday waves confined to a circular cavity. Its solutions are investigated theoretically and numerically. The horizontal trajectory of the droplet can be chaotic: it then corresponds to a random walk of average step size equal to half the Faraday wavelength. An analogy is made between the diffusion coefficient of this random walk and the action per unit mass h/m of a quantum particle. The statistics of droplet position and speed are shaped by the cavity eigenmodes, in remarkable agreement with the solution of Schrödinger equation for a quantum particle in a similar potential well.

Full text

PHYSICAL REVIEW E 93, 042202 (2016)
Quantumlike statistics of deterministic wave-particle interactions in a circular cavity
Tristan Gilet*
Microfluidics Lab, Department of Aerospace and Mechanics, University of Li`
ege, B-4000 Li`
ege, Belgium
(Received 21 January 2016; published 5 April 2016)
A deterministic low-dimensional iterated map is proposed here to describe the interaction between a bouncing
droplet and Faraday waves confined to a circular cavity. Its solutions are investigated theoretically and numerically.
The horizontal trajectory of the droplet can be chaotic: it then corresponds to a random walk of average step size
equal to half the Faraday wavelength. An analogy is made between the diffusion coefficient of this random walk
and the action per unit mass /m of a quantum particle. The statistics of droplet position and speed are shaped
by the cavity eigenmodes, in remarkable agreement with the solution of Schr¨
odinger equation for a quantum
particle in a similar potential well.
DOI: 10.1103/PhysRevE.93.042202
I. INTRODUCTION
The interaction between Faraday waves and millimeter-
sized bouncing droplets (Fig. 1) has attracted much atten-
tion over the past decade owing to its reminiscence of
quantum-particle behavior [1,2]. Faraday waves are stationary
capillary waves at the surface of a vertically vibrated liquid
bath [3–5]. They appear spontaneously and sustainably when
the vibration amplitude exceeds the Faraday threshold. At
smaller amplitude, they only appear in response to a finite
perturbation (e.g., the impact of a bouncing droplet) and
are then exponentially damped [6]. Close to threshold the
Faraday instability is often subharmonic, i.e., the Faraday wave
frequency is half the frequency of the external vibration [4].
The selected Faraday wavelength λFis approximately given
by the dispersion relation of water waves. The memory Mis
defined as the decay time of Faraday waves below threshold,
divided by the wave period; it increases and diverges as the
threshold is approached.
Liquid droplets are able to bounce several successive times
on liquid interfaces before merging, and this in a wide range
of conditions [7–9]. Rebounds can be sustained by vertically
vibrating the liquid interface [10–14]. When the rebound
dynamics locks into a periodic state with one impact every two
forcing periods, the droplet becomes a synchronous emitter of
Faraday waves [15]. More exactly, the droplet creates a radially
propagating circular capillary wave that excites standing
Faraday waves in its wake [6]. The resulting wave field then
contains contributions from the last Mdroplet impacts. In
the walking state, horizontal momentum is transferred from
the wave field to the impacting droplet, proportionally to the
local wave slope at the droplet position. The waves only exist
in response to droplet impacts, and in turn they cause the
horizontal motion of the droplet. This coupled wave-particle
entity at the millimeter scale is called a walker.
The dynamics of individual walkers has been investigated
experimentally in several configurations where it has shown
properties reminiscent of quantum particles [1]. For example,
walkers tunnel through weak boundaries [16]. They diffract or
interfere when individually passing through one or two slits,
respectively [17] (although these experimental results have
*Tristan.Gilet@ulg.ac.be
recently been challenged by theoretical arguments [18,19]).
Walkers also experience quantized orbits in response to
confinement by either central forces [20], Coriolis forces
[21–23], or geometry [24–26]. In the latter case, the walker
evolves in a cavity of finite horizontal extent. Chaotic tra-
jectories are often observed in the high-memory limit, when
confinement compels the walker to cross its own path again
after less than Mimpacts. In response to central and Coriolis
forces, the walker oscillates intermittently between several
different quantized orbits, as if it were in a superposition
of trajectory eigenstates [22,27,28]. The spatial extent of
these eigenstates is always close to an integer multiple of
half the Faraday wavelength λF/2, which can then be seen
as the analog of the de Broglie wavelength for quantum
particles [21]. The probability to find the walker in a given
state is proportional to the relative amount of time spent in
this state. In cavity experiments [24], the probability to find
the walker at a given position is strongly shaped by the cavity
geometry, as would be the statistical behavior of a quantum
particle in a potential well.
Several theoretical models have already been proposed
that capture various aspects of walker dynamics [6,25,28–31].
Most of them consider a stroboscopic point of view, where the
droplet is assumed to impact the bath perfectly periodically.
The wave field is then expressed as a sum of contributions from
each successive impact [6]. The walker horizontal trajectory
is deduced from Newton’s second law, where the driving force
is proportional to the local wave slope and the effective mass
includes both the real droplet mass and an added mass from the
wave [32]. The resulting discrete iterated map can be turned
into an integrodifferential equation for the walker trajectory in
the limit of small horizontal displacements between successive
impacts [29]. In an infinite space, the contribution from one
droplet impact to the wave field is assumed to be a Bessel
function of the first kind J0[kF(x−xd)] centered on the droplet
position xd, and of wave number kF=2π/λF[6]. The recent
model of Milewski et al. [31] provides a more sophisticated
model of the wave fields through inclusion of weak viscous
effects. It also includes realistic models of vertical bouncing
dynamics [13,33]. At the other end of the scale of complexity,
a generic model [30] has been proposed that reproduces
some key features of confined wave-particle coupling and
walker dynamics within a minimal mathematical framework.
These features include the time decomposition of the chaotic
2470-0045/2016/93(4)/042202(15) 042202-1 ©2016 American Physical Society

TRISTAN GILET PHYSICAL REVIEW E 93, 042202 (2016)
FIG. 1. (top) A drop of silicone oil bounces periodically at the
surface of a small pool (diameter 1.5 mm) of the same liquid which
is vertically vibrated at 80 Hz. (bottom) Drops of appropriate size
(here, diameter of 0.74 mm) can generate and interact with underlying
Faraday waves. This wave-particle association is called a walker. The
bottom left (right) picture shows a walker at low (high) memory, i.e.,
far from (close to) the Faraday-instability threshold.
trajectory into eigenstates [28], and the particle statistics being
shaped by confinement [24].
These hydrodynamic experiments and models have already
revealed a strong analogy between the statistical behavior
of chaotic walkers and quantum particles, in many different
configurations. It certainly results from the deterministic chaos
inherent to such wave-particle coupling. Nevertheless, it is
still unclear which ingredients are actually necessary and
sufficient for these quantum behaviors to appear. Also, no
direct connection between the equations of motion of the
walker and the Schr¨
odinger equation has been made yet. What
would be the equivalent of Planck’s constant for walkers? On
which timescale should the walker dynamics be averaged to
recover quantumlike statistics? This work aims at answering
these questions through a theoretical investigation of walker
dynamics under confinement in a two-dimensional cavity
[24].
First, the model developed in Ref. [30] is particularized
to a walker in a circular cavity (Sec. II). In Sec. III the
walker dynamics is analyzed as a function of its memory,
and results are compared to the experiments of Harris
et al. [24]. In Sec. IV it is shown that the present model gives
theoretical access to the experimentally unachievable limit of
zero damping (infinite memory) and perfect mode selection.
Finally, the above questions are addressed through direct com-
parison with analytical predictions from quantum mechanics
(Sec. V).
II. WAVE-PARTICLE COUPLING IN A CONFINED
GEOMETRY
The generic model [30] of the interaction between a particle
and a stationary wave confined to a domain Sis first recalled,
here in a dimensional form. It is primarily based on the
decomposition of the standing Faraday wave field Hn(X)
resulting from the nfirst impacts into the discrete basis of
eigenmodes k(X) of the domain:
Hn(X)=
k
Wk,nk(X),W
k,n =S
Hn(X)∗
k(X)dS, (1)
where ∗
kis the conjugate of k. These eigenmodes are
orthonormal on S. They are here chosen to satisfy the Neumann
condition n·∇k=0 at the boundary ∂S with normal
vector n.
We consider the stroboscopic approach where the droplet
impacts the liquid bath at regular time intervals. At rebound
n, it impacts at position Xnand creates a crater of shape Z=
F(R) in the wave field. In the limit of a droplet size much
smaller than λF, the crater F(R) can be approximated by a δ
function weighted by the volume of liquid displaced [30]:
F(R)=
k
∗
k(Xn)k(R)=δ(R−Xn).(2)
The contribution ∗
k(Xn) of the impact to each wave
eigenmode depends on the droplet position.
Each mode k(X) is given a viscous damping factor μk∈
[0,1] that depends on the forcing amplitude. It is defined as
the amplitude of mode kright before impact n+1, divided by
its amplitude right after impact n. The associated memory Mk
is given by Mk=−1/ln μk. The Faraday threshold instability
corresponds to max(μk)=1. The amplitude of each mode
Wk,n then satisfies the recurrence relation
Wk,n+1=μk[∗
k(Xn)+Wk,n]
=
n

n=0
μn+1−n
k∗
k(Xn).(3)
At each impact, the particle is shifted proportionally to the
gradient of the wave field at the impact position:
Xn+1−Xn=−δ
k
Wk,n ∇k]Xn
=−δ 
k
n−1

n=0
μn−n∗
k]Xn∇k]Xn,(4)
where δ>0 is the proportionality constant between the local
wave slope and the particle displacement. Equations (3) and (4)
form an iterated map that describes the evolution of the particle
in a cavity of arbitrary shape and dimension.
A. The circular cavity
Focus is now made on a two-dimensional circular cavity
of radius Rc, as studied experimentally by Harris et al. [24].
Cavity eigenfunctions are
k =φk
Rc
,φ
k(r,θ )=ϕk(r)eikθ ,k∈Z,∈N0,
(5)
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QUANTUMLIKE STATISTICS OF DETERMINISTIC WAVE- . . . PHYSICAL REVIEW E 93, 042202 (2016)
TABLE I. Twelve dominant Neumann eigenmodes for a cavity
of radius 14.3 mm filled with 20 cS oil and forced at 83 Hz.
Mode k, ΛFμ M Mode k, ΛFμ M
2,6 0.31 0.999 738 9,3 0.33 0.983 60
0,7 0.32 0.998 460 18,1 0.34 0.974 39
4,5 0.34 0.993 141 10,3 0.32 0.968 32
13,2 0.33 0.990 99 12,2 0.34 0.962 26
17,1 0.33 0.989 90 6,4 0.34 0.951 20
7,4 0.32 0.987 77 5,5 0.32 0.933 15
where r=R/Rcis the dimensionless radial position, θis the
angular position, and φk are the dimensionless eigenfunctions.
The dimensionless Faraday wavelength is defined as F=
λF/Rc. The radial functions ϕk are expressed as
ϕk(r)=1
π1−k2
z2
k Jk(zkr)
Jk(zk),(6)
where zk is the th zero of the derivative of the Bessel function
(first kind) of order k,soJ
k(zk)=0. The integer kcan take
both negative and positive values. The eigenfunctions satisfy
orthonormality conditions
2π
0
dθ 1
0
φkφ∗
krdr =δkkδ(7)
as well as the Neumann boundary condition ∂rφk =0inr=
1. Some of these eigenmodes are illustrated in Table I.
The dimensionless Faraday wave field hn(r,θ )=HnR2
c/
can be decomposed as a Dini series in this basis:
hn(r,θ )=+∞

k=−∞
∞

=1
wk,nφk(r,θ ),(8)
with
wk,n =Rc
Wk,n =2π
0
dθ 1
0
hn(r,θ )φ∗
krdr. (9)
The condition w−k,n =w∗
k,n results from hn(r,θ ) being real-
valued. Therefore,
hn(r,θ )=+∞

k=0
∞

=1
ξkRe[wk,nφk(r,θ )],(10)
where ξk=2 when k>0 and ξk=1 when k=0. For the sake
of readability, we remove the factor ξkand index from most
notations, keeping in mind that, when cylindrical harmonics
are involved, functions that depend on kalso depend on .
Specifically, the abbreviated sum over kwill always refer to
a double sum over kand ,from0to∞and from 1 to ∞,
respectively, with inclusion of the factor ξk.
The particle position is expressed in cylindrical coor-
dinates Xn/Rc=(rncos θn,rnsin θn). The wave recurrence
relation (3) then becomes
wk,n+1=μk[wk,n +ϕk(rn)e−ikθn].(11)
The trajectory equation (4) can be projected along Xnthen in
the direction perpendicular to Xn:
rn+1cos (θn+1−θn)=rn−C∂hn
∂r (rn,θn)
,
(12)
rn+1sin (θn+1−θn)=−C
rn∂hn
∂θ (rn,θn)
,
where
C=δ
R4
c
(13)
is a dimensionless constant that characterizes the intensity of
the wave-particle coupling. After calculating the gradient of
hnat the impact point, the iterated map becomes
rn+1cos (θn+1−θn)=rn−C
k
ϕ
k(rn)Re[wk,neikθn],
rn+1sin (θn+1−θn)=C
rn
k
kϕk(rn)Im[wk,neikθn],(14)
wk,n+1=μk[wk,n +ϕk(rn)e−ikθn],
where ϕ(r)=dϕ/dr.
B. Model discussion
This model of walkers neglects traveling capillary waves,
similarly to most previous works [6]. Indeed, these waves are
not re-energized by the vertical forcing, and their initial energy
spreads in two dimensions. Their amplitude is then smaller
than Faraday waves, and their contribution is likely to have
only marginal influence on the walker’s long-term behavior.
Only when the walker comes close to the boundary could these
capillary waves significantly modify the local trajectory. But
this effect would then be localized in space and time, so we
assume that it does not strongly affect the walker statistics.
In contrast to previous models [29,31], the trajectory
equation (4) is here first order in time. It therefore assumes that
the walker completely forgets its past velocity at each impact,
i.e., horizontal momentum is entirely dissipated. Nevertheless,
an inertial effect is still present, because the walker keeps a
constant velocity between successive impacts. An additional
inertial term, e.g., corresponding to the hydrodynamic boost
factor [32], would increase the model complexity and the
number of parameters, so it is left to future work.
This model should be compared to the experimental results
of a walker in a circular cavity reported by Harris et al. [24]. A
necessary step to a quantitative comparison is the accurate
determination of each eigenmode kand its associated
damping factor μk. Unfortunately, these modes are not easily
characterized experimentally, since they cannot be excited
one at a time. Uncertainty in the boundary conditions further
complicates mode identification. Ideal Dirichlet boundary
conditions (zero velocity where the liquid-air interface meets
the solid walls) could have been expected if the contact line
was pinned [34]. But in experiments contact lines are avoided
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TRISTAN GILET PHYSICAL REVIEW E 93, 042202 (2016)
on purpose because the vibration of the associated meniscus
would be a source of parasitic capillary waves. Instead, the
solid obstacles that provide confinement are always slightly
submerged [16,17,24,26]. The walker cannot penetrate these
shallow regions since waves are strongly dissipated there [16].
Recent numerical simulations suggest that such boundary
condition cannot be strictly expressed as a Robin condition
[35]. From a practical point of view, when Dirichlet boundary
conditions are considered in Eq. (14), the particle tends to
leave the cavity as its radial displacement does not vanish at
the boundary. For this reason, Neumann boundary conditions
(zero wave slope) are adopted in this work.
Determining the damping factor associated with each mode
is also an issue. In the absence of horizontal confinement,
viscous wave damping can be estimated by spectral meth-
ods [5,31]. The theoretical prediction is then validated by
measuring the Faraday threshold amplitude as a function
of forcing frequency. Unfortunately, confinement with sub-
merged vertical walls yields additional viscous dissipation.
Indeed, modes φk of high kand small are never observed
in experiments right above the Faraday threshold (D. Harris,
private communication), while spectral methods would predict
them as highly unstable.
III. FINITE MEMORY IN A DAMPED WORLD
Harris and coworkers [24] considered a circular cavity
of radius Rc=14.3 mm filled with silicone oil of density
965 kg/m3, surface tension σ=20 mN/m, and kinematic
viscosity ν=20 cS. The forcing frequency f=70 Hz was
chosen in relation with the cavity size, such that the most
unstable mode at Faraday threshold is one of the radial
modes (k=0, = 0). In this work, damping factors are
approximately determined from a spectral method [5] for this
same cavity with Neumann boundary conditions and various
forcing frequencies. A frequency of 83 Hz is then selected
that again gives predominance to one of the purely radial
modes. Only modes of damping factor μk>0.01 are retained.
The twelve modes of highest damping factor are illustrated in
Table I. Their wavelength is always very close to the Faraday
wavelength that would result from this forcing frequency in
the absence of confinement.
The average walking speed vwincreases with the coupling
constant C. In experiments [24], vw=8.66 mm/s at a forcing
frequency of f=70 Hz and at 99% of the Faraday threshold.
This corresponds to an average step size 2vw/f =0.017Rc.
The same value is obtained when Eq. (14) is solved with
C=3×10−5. If we further assume that δ∼2vw/f ,the
volume of fluid displaced at each impact can be estimated
from Eq. (13)tobe∼5μL. This volume corresponds to
an interface deflection of a few hundred micrometers on a
horizontal length scale of a few millimeters. It is one order
of magnitude higher than the droplet volume, even when
corrected with the boost factor [32].
A. Circular orbits
The iterated map (14) admits periodic solutions (Fig. 2)
where the particle orbits at constant speed around the center of
the cavity: rn=r1,θn=αn, and wk,n =wk1e−ikαn (the wave
pattern also rotates). The walking velocity is r1α. Plugging
FIG. 2. Convergence to a stable circular orbit of type A (r1=
0.592, α=0.0135) at memory M=7.7andC=3×10−5.The
underlying wave-field corresponds to the time at which the walker is
at the position indicated by the black circle (•).
this solution into the iterated map yields
wk1=μk
e−ikα −μk
ϕk1,
r1(1−cos α)=C
k
ϕk1ϕ
k1μk
cos (kα)−μk
1−2μkcos (kα)+μ2
k
,
r2
1sin α=C
k
kϕ2
k1μk
sin (kα)
1−2μkcos (kα)+μ2
k
,(15)
where ϕk1=ϕk(r1) and ϕ
k1=[dϕk/dr](r1). A detailed analy-
sis of these periodic solutions is given in Appendix A.Atlow
memory, the only solution to Eq. (15)isα=0 (fixed points).
The corresponding radii r1satisfy
(r1)=0,with (r)=
k
μk
1−μk
ϕ2
k(r)
2.(16)
As memory increases (i.e., as all μkincrease), each of these
fixed points experiences a pitchfork bifurcation where two
other solutions ±α= 0 appear that correspond to clockwise
and counterclockwise orbits. This bifurcation is analogous to
the walking threshold observed and rationalized for unconfined
walkers [6,29]. These orbits emerge from the loss of stability
of their corresponding fixed point. The radial stability of
both fixed points and corresponding orbits is related to
(r), as already shown in Ref. [30] for the unimodal and
unidimensional version of Eqs. (3) and (4). They are usually
stable when (r1)>0 (orbits of type A) and unstable when
(r1)<0 (orbits of type B). For single-frequency forcing,
successive orbits of a same type are approximately separated
by F/2. The convergence of trajectories towards stable orbits
of type A involves wobbling, i.e., radial oscillations (Fig. 2).
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QUANTUMLIKE STATISTICS OF DETERMINISTIC WAVE- . . . PHYSICAL REVIEW E 93, 042202 (2016)
r
0 0.2 0.4 0.6 0.8 1
M
5
6
7
8
9
10
15
20
30
50
100
FIG. 3. Bifurcation diagram of the radial position ras a function
of memory Mat C=3×10−5. The probability distribution function
is represented in levels of gray; darker regions correspond to more
frequently visited radial positions. Thick solid lines and thin dashed
lines correspond to stable and unstable fixed points, respectively
[Eq. (16)]. Full and empty symbols represent stable and unstable
circular orbits, respectively [Eq. (15)]. Circles (◦) indicate that
at least one pair of eigenvalues is complex-conjugated (type A)
while triangles () are used when all corresponding eigenvalues are
real-valued (type B).
B. Transition to chaos
As memory increases, each circular orbit of type A
destabilizes radially through a Neimark–Sacker bifurcation
(Fig. 3)[30]. Orbits of larger radius destabilize at higher
memory, since it takes more rebounds before the particle
revisits its past positions. Sustained wobbling orbits of finite
amplitude are then observed, where the radial position oscil-
lates periodically. As memory is increased further, wobbling
orbits progressively disappear. At high memory (M>50 in
Fig. 3), there are no stable periodic attractors left and the walker
chaotically explores the entire cavity. The chaotic nature of the
system is highlighted by the exponential growth of the distance
between two trajectories that were initially extremely close to
one other (positive Lyapunov exponent in Fig. 4).
A chaotic trajectory is represented in Fig. 5. Several
trajectory patterns are recurrent, including abrupt changes in
direction and off-centered loops of characteristic radius close
to half the Faraday wavelength F/2. Overall, the walker is
observed to spend significantly less time at radial positions
where orbits were stable at low memory (Fig. 3).
IV. ACCESSIBLE INFINITE MEMORY
Unlike its hydrodynamic analog, a quantum particle con-
fined in a cavity is known to be a Hamiltonian system for which
there is no dissipation. The behavior of individual walkers that
are reminiscent of quantum particles were always observed
at high memory, i.e., when dissipation is almost balanced
by external forcing. Indeed, in these conditions, the walker
gets a chance to walk onto the wave field left in its own
N
1000 1200 1400 1600 1800 2000
|r1-r2|
10-12
10-10
10-8
10-6
10-4
10-2
100
x
-0.5 0 0.5
y
-1.2
-1
-0.8
-0.6
-0.4
-0.2
FIG. 4. Exponential divergence of the radial distance |r1−r2|be-
tween two neighboring trajectories, initially separated by |r1−r2|=
10−10 and placed on the same initial wave-field. Memory and coupling
constants are M=500 and C=3×10−5, respectively. The inset
shows the two trajectories represented with a solid line and dots,
respectively, for N∈[1400,1600].
wake [20,22,36]. For each mode k, this balance is theoretically
achieved when μk=1 (infinite memory). Memory above 100
is very challenging to achieve experimentally, owing to the
imperfect control of the forcing vibration [37]. Moreover, in
most geometries (including circular), cavities are such that
it is impossible to get several modes reaching μk=1for
the same forcing acceleration. This model does not suffer
from such limitations; it is possible to select a subset Sof
x
-1 -0.5 0 0.5 1
y
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
FIG. 5. Chaotic trajectory at memory M=500 and C=3×
10−5. The solid line corresponds to 10 000 impacts, while the dots
emphasize 380 of these successive impacts.
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TRISTAN GILET PHYSICAL REVIEW E 93, 042202 (2016)
TABLE II. Subsets of selected modes S, corresponding range of
wavelength and coupling constant C, and associated symbols and
colors in subsequent figures.
SSelected mo des (k, )∈S ΛFΔΛFlog10 C
S1–(0,5); (2,4); (6,3); 0.473 4% -5
(9,2); (13,1)˝
S2S1∪–(3,4); (5,3); 0.473 8% -6.5
(7,3);(10,2);(14,1) ˝ -6
-5.5
-5
-4.5
-4
-3.5
-3
-2.5
S3S2∪–(1,5); (1,4); (4,4); 0.473 16% -5
(4,3); (8,3); (8,2);
(11,2); (12,1); (15,1) ˝
S4–(0,12); (2,11); (7,9); 0.178 1% -6
(10,8); (13,7); (17,6);
(26,3); (30,2); (35,1) ˝
•S5–(0,31); (2,30) ; (11,26) ; 0.066 0.2% -7
(24,21); (27,20); (47,13);
(62,8); (69,6); (77,4) ˝
S6S1∪–(0,12); (2,11); 0.473 4% -5.3
(7,9); (13,7) ; (17,6) ˝ + 0.178 0.5%
modes for which the memory is infinite (μk=1,k ∈S) while
all other modes are instantaneously damped (zero memory,
μk=0,k /∈S). In the following, several subsets of selected
modes Sare considered (Table II). Each subset includes all the
modes for which the characteristic wavelength lies in a narrow
range F[1 −F,1+F], except the latest subset which
is a combination of two wavelengths. Such mode selection
could be analog to preparing quantum particles with localized
momentum. For subset S2the coupling constant Cis varied
over four orders of magnitude.
A. Radial statistics
Figure 6shows a superposition of half a million impact po-
sitions. Dark circular strips indicate more frequent impacts at
certain radial positions r, as observed in experiments [24]. The
corresponding probability distribution function ρ(r)showsa
series of extrema at radii that are largely independent of C
[Fig. 7(a)]. Chaotic trajectories in the infinite-memory limit
are qualitatively identical to those arising at finite memory.
The radial position oscillates intermittently between several
preferred radii that correspond to the maxima of ρ(r)[30]
[Fig. 7(b)].
The iterated map is checked to be ergodic to a good
approximation (within 1% in these simulations), so the time-
average of any property over a single trajectory coincides with
an ensemble average of this property over many independent
FIG. 6. Two-dimensional probability distribution function ob-
tained by superimposing half a million impact positions issued from
78 independent trajectories. The subset of selected modes is S2
(Table II), with C=10−5.
trajectories at a given time. In other words,
lim
N→∞
1
N
n+N

n=n
f(rn,θn)=2π
01
0
ρ(r)f(r,θ )rdrdθ, (17)
for any regular function f(rn,θn). It is rather counterintuitive
that the wave-field does not blow up with time, since contri-
butions from previous impacts keep adding up without any of
them being damped out. The resulting wave field amplitudes
satisfy
wk,n =
n−1

n=0
ϕk(rn)e−ikθn.(18)
In the long term, ergodicity yields
lim
n→∞ wk,n =n2π
0
e−ikθdθ 1
0
ρ(r)ϕk(r)rdr. (19)
The angular integral vanishes by symmetry, except when
k=0. In this latter case, the only way to keep a finite wave
amplitude is to satisfy
1
0
ρ(r)ϕ0(r)rdr =0.(20)
So ρ(r) must always be orthogonal to each of the radial
eigenmodes φ0(r). We have checked numerically that for every
subset of selected modes S(Table II), the component of ρ(r)
along φ0(r) is always at least two orders of magnitude smaller
than the largest other component.
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QUANTUMLIKE STATISTICS OF DETERMINISTIC WAVE- . . . PHYSICAL REVIEW E 93, 042202 (2016)
ρ(r)
0 0.2 0.4 0.6 0.8 1
r
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a)
n
0 200 400 600 800 1000
rn
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b)
FIG. 7. (a) Probability distribution function ρ(r) of radial position rfor mode subset S2and three different values of C(Table II). The solid
line corresponds to the quantum prediction given in Eq. (35). (b) An example of the time evolution of the particle’s radial position for mode
subset S2and for C=10−5.
B. Coherence and diffusion
The walker trajectory is usually smooth and regular at the
scale of a few impacts (Fig. 5). But such coherence is lost
on longer timescales, where the trajectory oscillates and loops
chaotically. This behavior is quantified through the average
distance d(n)traveledinnsteps, defined as
d(n)=||xn+n−xn||2n,(21)
and represented in Fig. 8. For small nthis distance increases
linearly with nso the motion is ballistic (Fig. 8). As soon
as dF/2, dstarts increasing proportionally to √n, like
in normal diffusion. It then saturates in d2 owing to the
n / Nc
10-2 10-1 100101102103
d / ΛF
10-2
10-1
100
101
n / Nc
0.01 1 100
d / ΛF
0.01
0.1
1
FIG. 8. Average distance dtraveled in nsuccessive impacts, for
subsets S1,S4,S5and S6. The solid line corresponds to the ballistic
regime d/F=an/Nc, while the dashed line corresponds to the
diffusive regime d/F=b√n/Nc. (Inset) Same plot, for S1,S2and
S3. Symbols and colors are explained in Table II.
finite size of the cavity. This ballistic-to-diffusive transition
was already observed for the one-dimensional (1D) version
of this model [30]. This suggests that the walker trajectory is
similar to a random walk for which the elementary steps are
of the order of F/2.
The average number of impacts in one step should depend
on the walker velocity. It can be estimated by first looking
at the amplitude wk,n of each wave eigenmode (see inset of
Fig. 9). Their time evolution is a succession of coherent linear
r
0 0.2 0.4 0.6 0.8 1
<|wk,n+1-wk,n|>
0
0.5
1
1.5
2
2.5
3
n
0 100 200 300 400
Re(wk,n)
-40
-20
0
20
40
Nw
FIG. 9. Average increment of wave amplitude at each impact
|wk,n+1−wk,n| as a function of particle position r, for modes of the
subset S2. The solid line is |ϕk(r)|. Modes (k,)=(0,5) and (6,3) are
represented in black and gray, respectively. Inset shows an example
of the time evolution of the wave amplitude Re(wk,n), for these same
modes. Nwis defined as the distance between two successive extrema
of wk,n.
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TRISTAN GILET PHYSICAL REVIEW E 93, 042202 (2016)
Nw
0 50 100 150 200
PDF
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Nc
1 10 100 1000
<Nw>
1
10
100
1000
ΛF/2
0 0.1 0.2 0.3
Lw
0
0.1
0.2
0.3
FIG. 10. Probability distribution function of the coherence time
Nwfor the two modes (k,)=(0,5) and (6,3) of subset S2(in black
and gray, respectively). The vertical line corresponds to Nw=Ncas
defined in Eq. (23). Top inset shows average distance Lwtraveled
during a coherent segment Nwas a function of the half Faraday
wavelength F/2. Bottom inset shows average number of impacts
Nwof a coherent segment as a function of Nc. In both insets, data
from all subsets are represented with different symbols (Table II), and
the solid line corresponds to equal abscissa and ordinate.
segments of variable duration Nw. This constant growth rate
depends on the particle position; at every impact, each wk
increases by an average amount
|wk,n+1−wk,n|n|ϕk(r)|,(22)
as confirmed in Fig. 9. The statistical distribution of Nw
peaks at approximately the same value for each mode of a
given selection, then it decreases exponentially for large Nw
(Fig. 10). So after the average coherence time Nweach
mode amplitude wkhas grown by about Nw|ϕk|and its
contribution to the total wave slope now scales as ξkwk∇ϕk∼
2ξkNcϕ2
k/F. The average displacement per impact is then
x ∼2CχNwϕ2
k/(F), where χ=kξk. The average
distance Lw=[||xn+Nw−xn||2n]1/2traveled during one of
these coherent segments is observed to be close to F/2,
independently of Cor F(see upper inset in Fig. 10).
We deduce that Nw∼F(4Cχϕ2
k)−1/2, which yields the
definition of the coherence timescale
NcF
√Cχ =λFRc
√δχ .(23)
The bottom inset of Fig. 10 validates the scaling law Nw
Ncover four decades of C,forseveralFand χ, with a
proportionality constant almost equal to unity. Even the subset
S6of mixed wavelengths () satisfies these scaling arguments.
The curves of average distance vs time perfectly collapse
on a single curve independent of F(or χ) and C, when n
and dare normalized by Ncand F, respectively (see inset
of Fig. 8). Only subsets corresponding to Nc<4 (i.e., the
two largest values of Cfor S2) fail to collapse perfectly. The
ballistic regime is described by
d
F=an
Nc
,(24)
where a0.57. Similarly the diffusive regime satisfies
d
F=bn
Nc
,(25)
where b0.62 based on the data of subset S5, for which the
diffusive region is the largest since Fis the smallest (Fig. 8).
If the diffusive behavior is attributed to a two-dimensional (2D)
random walk, then the corresponding diffusion coefficient is
defined as
˜
Dd2
4n=b2
4
2
F
Nc
.(26)
The crossover between both regimes occurs in n/Nc=
b2/a21.2, so in d/F=b2/a 0.67. It can be seen as the
elementary step of this random walk, and it is slightly larger
than half the Faraday wavelength. In dimensional terms, the
diffusion coefficient is
D=R2
cf˜
D=b2
4
λ2
Ff
Nc0.096 λ2
Ff
Nc
,(27)
where fis the impact frequency.
V. COMPARISON WITH PREDICTIONS OF QUANTUM
MECHANICS
In this section, the analogy between walkers at infinite
memory and quantum particles is further developed. More
exactly, the solutions of the map (14) are compared to the
predictions of the Schr¨
odinger equation for a quantum particle
subject to the same confinement.
A. Correspondence of timescales
A single quantum particle is statistically described by a
wave function (x,t): The probability to find the particle
at position xis given by ρ(x,t)=||2. For nonrelativistic
free particles, the wave function evolves according to the
Schr¨
odinger equation
i∂t+
2
2m∇2=0,(28)
where is the reduced Planck constant and mis the rest mass
of the particle. Special relativity can be taken into account by
instead considering the relativistic Klein–Gordon equation for
the function ψ(x,t), which applies for spinless particles:
1
c2∂ttψ+m2c2
2ψ−∇2ψ=0.(29)
In his pilot-wave theory, de Broglie [38] hypothesized
that quantum particles would vibrate at the Compton fre-
quency ωc=mc2/(Zitterbewegung). The nonrelativistic
Schr¨
odinger equation can be retrieved by expressing ψas the
modulation of a nonrelativistic, slowly varying wave function
(x,t) by this vibration at the Compton frequency:
ψ(x,t)=e−imc2
t(x,t).(30)
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QUANTUMLIKE STATISTICS OF DETERMINISTIC WAVE- . . . PHYSICAL REVIEW E 93, 042202 (2016)
Substitution in Eq. (29) indeed yields
i∂t+
2
2m∇2=
2
2mc2∂tt. (31)
The left-hand side is identical to Eq. (28), so the right-hand
side is a relativistic correction. The characteristic frequency of
(x,t)isω=k2/(2m). Therefore, the relativistic correction
is negligible when ω/k c, or equivalently when kλc1,
where λc=2π/(mc) is the Compton wavelength.
Couder and coworkers have already observed in many dif-
ferent configurations [17,20,21] that the Faraday wavelength
λFwas the walker equivalent of the de Broglie wavelength
λdB =2π/k of quantum particles (Analogy No. 1). In a recent
review paper, Bush [1] pushed the analogy one step further and
hypothesized that the bouncing motion of walkers could be the
equivalent of this Zitterbewegung, so the bouncing frequency
fwould correspond to the Compton frequency mc2/(2π)
(Analogy No. 2).
The diffusive behavior of the walker on the long term
suggests a third analogy (No. 3), between the walker’s
diffusion coefficient Dand the coefficient of Schr¨
odinger
equation /m. It is equivalent to say that the Schr¨
odinger
timescale 2π/ω =4πm/(k2) is analogous to the dimensional
coherence time of the walker λ2
F/(πD)[4/(πb2)]Ncf−1
3.3Ncf−1. De Broglie’s momentum relation states that the
speed of a nonrelativistic particle is v=k/m, which is
then equivalent to [πb2/2]λFfN−1
c0.60λFfN−1
cfor the
walker. This value is remarkably close to the ballistic speed
aλFf/Ncwith a=0.57 as observed in Fig. 8. The analog of
the speed of light cis [b√π/2]λFfN−1/2
c=0.78λFfN−1/2
c,
which does not seem to be a fundamental constant in the
walker’s world. Nevertheless, the relativistic limit v→c
corresponds to Nc→πb2/20.60 for the walker, i.e., the
coherence time becomes of the order of the bouncing period.
Saying that a quantum particle cannot go faster than light
is then analogous to saying that the coherence time of a
walker trajectory should be at least one rebound time. When
Nc=πb2/20.60, the equivalent speed of light becomes
λFf, which is the phase speed of capillary waves. Finally,
in quantum mechanics the correspondence principle states
that classical mechanics is recovered in the limit →0. This
translates into the walker behavior being fully ballistic when
Dgoes to zero, i.e., when the coherence timescale Ncgoes
to infinity. The three equivalences and their implications are
summarized in Table III.
B. Probability density
In a 2D circular infinite potential well of radius Rc,thewave
function can be decomposed into a discrete basis of cylindrical
harmonics φk(r) defined in Eq. (5):
=1
Rc
k
ckφk(r)e−iωkt,(32)
where (R2
c∇2+z2
k)φk=0 and
ωk=

2mR2
c
z2
k(33)
in order to satisfy the Schr¨
odinger equation. The probability
density function (PDF) is then
ρ(r,t)=
j,k
c∗
jckφ∗
j(r)φk(r)e−i(ωk−ωj)t.(34)
Coefficients ckmust satisfy SρdS =k|ck|2=1. Since all
zkare distinct (there is no degeneracy), the time-averaged
probability density is
ρ(r)=
k|ckφk(r)|2=
k|ck|2ϕk(r)2,(35)
which is necessarily axisymmetric. Although all zeros of
Bessel functions (and derivatives) are strictly distinct, it is
always theoretically possible to find and select two eigenmodes
(j,k)forwhich|zj−zk|is arbitrarily small. The corre-
sponding imaginary exponential in Eq. (34) would generate
some beating at the extremely low frequency (ωj−ωk) that
would challenge any practical computation of a time average.
Nevertheless, as far as the modes selected in this work are
concerned (Table II), |zj−zk|>0.01 for any j= k.
How well are the walker statistics described by these
quantum predictions? Following de Broglie’s pilot-wave the-
ory [38,39], we assume an analogy between the quantum wave
function and the classical wave field of the walker, averaged
over the coherent timescale. In Sec. IV, the walker dynamics
was considered in configurations where a small number of
modes were equally excited while others were strictly not.
This suggests the identification of quantum coefficients ckas
ck=ξk
kξ2
k
,(36)
where again ξk=1ifk=0 and ξk=2 otherwise. As seen in
Fig. 7(a), the corresponding quantum prediction of the average
probability density [Eq. (35)] is reminiscent but not identical
to the one obtained from walker simulations. Nevertheless,
their extrema of density coincide almost perfectly.
C. Average kinetic energy
In quantum mechanics, the kinetic energy operator ˆ
T=
−2
2m∇2is Hermitian, and its expected value
T=|ˆ
T|=−
2
2mS
∗∇2dS =
2
2mR2
c
k|ck|2z2
k
(37)
is time-independent in a infinite potential well. A dimension-
less kinetic energy can then be defined based on the cavity
radius Rcand Compton frequency ωc/(2π):

T=2
mT4π2
R2
cω2
c=B2
k|ck|2z2
k,(38)
where B=2π/(mR2
cωc) is a dimensionless coefficient.
According to Table III,Bis equivalent to the dimensionless
diffusion coefficient ˜
Dof the walker, defined in Eqs. (26)
and (27). This quantum prediction is in remarkable agreement
with the observed average kinetic energy of simulated walkers,
for all considered mode subsets (Fig. 11). Only simulations at
large Cfail to match the prediction, possibly because they
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TRISTAN GILET PHYSICAL REVIEW E 93, 042202 (2016)
TABLE III. Correspondence of variables between walkers and quantum particles (b0.62).
Analogy Walker Quantum particle
No. 1 Faraday wavelength λFde Broglie wavelength 2π/k
No. 2 Bouncing frequency fZitterbewegung frequency mc2/(2π)
No. 3 Diffusion coefficient D=[b2/4]λ2
FfN−1
cSchr¨
odinger diffusion /m
Derived Coherence time [4/(πb2)]Ncf−1Schr¨
odinger timescale 4πm/(k2)
Ballistic speed v=[πb2/2]λFfN−1
cQuantum speed v=k/m
Maximum speed [b√π/2]λFfN−1/2
cSpeed of light c
Relativistic limit Nc>πb
2/2 Relativistic limit v/c < 1
Ballistic limit Nc→∞ Classical limit →0
would be in the relativistic regime (Nc<5). The energy of
the mixed-mode subset S6is also overestimated. The kinetic
energy of two interacting walkers was shown to be somehow
equivalent to the energy stored in the corresponding wave
field [40]. Nevertheless, as detailed in Appendix B,itishere
unclear if both energies are still equivalent for a single walker
confined in a cavity.
D. Position-dependent statistics
Radial- and azimuthal-velocity operators are defined by
projecting the momentum operator ˆ
P=−i∇along the radial
and azimuthal directions, respectively:
ˆ
Vr=1
mer·ˆ
P=−i

mRc
∂r,
ˆ
Vθ=1
meθ·ˆ
P=−i

mRc
1
r∂θ.(39)
On the one hand, the radial-velocity operator is not Hermitian,
so it should not be observable. On the other hand, the
tangential-velocity operator has a trivial expected value of
zero, by symmetry. The Heisenberg uncertainty principle states
that one cannot measure accurately and simultaneously both
T
q
uantum
10-6 10-4 10-2 100
Twalker
10-6
10-4
10-2
100
FIG. 11. Average dimensionless kinetic energy 
Tof the walker,
vs quantum prediction, with the equivalence B≡˜
D. Symbols
correspond to different mode subsets (Table II). The solid line is
the quantum prediction [Eq. (38)].
the position and momentum of a quantum particle. However,
it is possible to “weakly” measure a quantum particle, gaining
some information about its momentum without appreciably
disturbing it, so its position can be “strongly” measured
directly after [41]. The information obtained from individual
measurements is limited. But one can perform many trials,
then postselect particles that were observed at a given position
and calculate their associated average momentum. We here
propose to extend this concept of weak measurement to a
particle in a 2D circular cavity. We define the Hermitian
operators
ˆ
V2
r=ˆ
V†
r
ˆ
δ(r−R)
2πR
ˆ
Vrand ˆ
V2
θ=ˆ
V†
θ
ˆ
δ(r−R)
2πR
ˆ
Vθ,(40)
which are aimed to represent the squared radial and azimuthal
velocities at a given radial position R, respectively. Their
expected values
|ˆ
V2
r|=
mRc2S
∂r∗∂rδ(r−R)
2πR dS,
(41)
|ˆ
V2
θ|=
mRc2S
∂θ∗∂θδ(r−R)
2πR r−2dS,
correspond to the variance of each velocity component. They
are time-averaged,
|ˆ
V2
r|=
mRc2
k|ck|2[∂rϕk]2
r=R,(42)
|ˆ
V2
θ|=
mRc2
k|ck|2k2[ϕk]2
r=R,(43)
and made dimensionless,

v2
r= |ˆ
V2
r|4π2
R2
cω2
c=B2
k|ck|2[∂rϕk]2
r=R,
(44)

v2
θ= |ˆ
V2
θ|4π2
R2
cω2
c=B2
k|ck|2k2[ϕk]2
r=R.
Again, the equivalence between Band ˜
Dallows for a
direct comparison between walkers and quantum predictions.
The evolution of 
v2
rwith Ris very similar in both worlds,
although the proportionality coefficient between both (best fit)
varies from one subset to another (Fig. 12). The agreement
is even better for 
v2
θ, where the walker variance is almost
exactly twice the quantum prediction, at any radial position,
for most values of C(Fig. 13). This remarkable similarity
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QUANTUMLIKE STATISTICS OF DETERMINISTIC WAVE- . . . PHYSICAL REVIEW E 93, 042202 (2016)
R
0 0.2 0.4 0.6 0.8 1
v2
r/˜
D2
0
100
200
300
400
500
600
FIG. 12. Average variance of the radial velocity 
v2
rof the walker at
a given radial position r, normalized by ˜
D2, for subset S2(Table II).
The solid line corresponds to 3.16 times the quantum prediction
[Eq. (44); best fit].
holds for any of the markedly different functions 
v2
θ(R)at
each mode subset considered (Fig. 14). However, the quantum
calculation strongly underestimates both walker variances at
large C(relativistic regime).
These results demonstrate how much the velocity statistics
of the walker are shaped by the wave function in almost the
same way as the statistics of a quantum particle would be.
Moreover, they lead to an interpretation of the Heisenberg
uncertainty principle for walkers (slightly different from the
one proposed in Ref. [17]). The uncertainty in position can be
related to the coherence length and it is then of the order of
λF/2 (equivalently λdB/2). The uncertainty in speed is directly
R
0 0.2 0.4 0.6 0.8 1
v2
θ/˜
D2
0
20
40
60
80
100
120
140
160
180
200
FIG. 13. Average variance of the tangential velocity 
v2
θof the
walker at a given radial position r, normalized by ˜
D2, for subset
S2(Table II). The solid line corresponds to 2.0 times the quantum
prediction [Eq. (44); best fit].
R
0 0.2 0.4 0.6 0.8 1
0
100
200
300
0
5000
10000
0
500
1000
0
50
100
150
FIG. 14. Variance of the tangential velocity 
v2
θof the walker at
a given radial position r, normalized by ˜
D2, for subsets S1,S4,S5,
and S6, from top to bottom (Table II). The solid lines correspond
respectively to 1.0, 2.1, 2.2, and 2.0 times the quantum prediction
associated with each subset [Eq. (44); best fit].
given by the dimensional version of the standard deviation √v,
which scales as D/λF[equivalently /(mλdB)]. The product
of these uncertainties then scales as the diffusion coefficient
D, which is equivalent to /m. So the uncertainty principle is
recovered for walkers provided their dynamics is analyzed at
the scale of diffusion.
VI. DISCUSSION
The theoretical framework introduced in Sec. II is one
of the simplest mathematical representations of a particle
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TRISTAN GILET PHYSICAL REVIEW E 93, 042202 (2016)
coupled to a wave. It qualitatively captures the key features
observed in the experiments on confined walkers performed
by Harris et al. [24]. It reproduces both the circular orbits at
low memory and the chaotic trajectories at high memory. The
current limitation for a quantitative comparison originates in
the lack of experimental information about the damping rate
of each eigenmode. Nevertheless, this model allows for the
exploration of regimes that are not accessible experimentally.
Of particular interest is the possibility to set the memory
of each mode to either infinity (no damping) or zero (full
damping). Quantumlike behavior of individual walkers were
observed at high-memory, when the system was as close to
conservative as it can be. In this model, the mode selection
around one given wavelength can be seen as analogous to the
preparation of a quantum state with a more-or-less defined
momentum. Nevertheless, it must be noted that this walker
model is still highly dissipative since the nonselected modes
(the ones that do not resonate with the forcing) are immediately
damped.
The chaotic trajectories of confined walkers are ballistic
on the short term and diffusive on the long term. The
coherence distance, beyond which the ballistic behavior is
lost, corresponds to half the Faraday wavelength, as a careful
qualitative look at the experimental data [24] also confirms.
The Faraday wavelength is at the heart of most quantumlike
behaviors of walkers; it is identified as equivalent to the de
Broglie wavelength for a quantum particle. Our analysis of
the diffusive motion in a circular corral has suggested another
equivalence, between the diffusion coefficient Dand the factor
/m in the Schr¨
odinger equation. The walker behavior thus
becomes apparently random only when it is analyzed at a
length scale larger than λF/2. This analogy is confirmed by
the observed ballistic speed of the walker, which corresponds
closely to the de Broglie speed k/m. Similarly, the average
kinetic energy of the walker matches Schr¨
odinger’s prediction
over several orders of magnitude.
The bouncing dynamics of the walkers was previously
hypothesized as reminiscent of the Zitterbewegung of quantum
particles [1]. From there, we found an analog for the speed of
light in the walker’s world, and we then identified the condition
for observing relativistic effects on walkers: they have to
travel a distance comparable to half the Faraday wavelength
at every rebound. Our mathematical framework allows for
a future investigation of this regime, which is unfortunately
not attainable in experiments where the walking steps are
usually limited to around λF/20 [6,24]. Equivalence relations
of Table III do not explicitly depend on the dispersion relation
of the waves. Nevertheless, the variables therein (such as the
coherence time or the elementary diffusion coefficient) do
depend on the considered wave-particle interaction, e.g., here
through the coupling constant C[Eq. (23)]. This might be the
reason why the analogs for the speed of light and the Planck
constant are not constant in the walker’s world.
Harris et al. [24] observed that the statistics of confined
walkers can be shaped by the cavity eigenmodes in the
high-memory limit. We have here calculated the position-
dependent variance of walker velocity in the limit of some
modes having an infinite memory. Their dependence on radial
position is remarkably close to the predictions obtained from
the quantum formalism (linear Schr¨
odinger equation) with
Hermitian observables, even when complex combinations of
modes are considered.
This model of walkers is reminiscent of the pilot-wave
theories of de Broglie and Bohm, although there are some
significant differences [1]. In the Bohmian mechanical de-
scription of quantum mechanics, particles are guided by
a pilot-wave prescribed by Schr¨
odinger’s equation. It thus
evolves at the Schr¨
odinger timescale. The particles do not
exert any direct individual feedback on this wave; only their
statistics shapes the wave. By contrast, individual bouncing
walkers locally excite the wave field that sets them into motion.
The double-solution theory of de Broglie [38]involvesan
additional pilot-wave centered on the particle, whose timescale
would be the Zitterbewegung period. This second wave could
be the analog of the real Faraday wave that couples with
individual walkers. It was already shown that the Schr¨
odinger
equation can be retrieved from a random walk of diffusion
coefficient /(2m)[42]. This work suggests that such a random
walk can originate from the chaos of a deterministic map that
describes the coupling of a wave and a particle. In other words,
the solution of Schr¨
odinger equation for a particle in a cavity
can be obtained from a purely deterministic mechanism that
does not involve any stochastic element.
Walkers have now been investigated for a decade. They have
shown many behaviors reminiscent of quantum particles. We
have shown here that, when they are confined in cavities, their
statistics closely approaches the solution of the Schr¨
odinger
equation. Future work is still required to identify the exact
limits of this analogy.
ACKNOWLEDGMENTS
This research was performed in the framework of the
Quandrops project, financially supported by the Actions de
Recherches Concertees (ARC) of the Federation Wallonie-
Bruxelles under Contract No. 12-17/02. This research has also
been funded by the Interuniversity Attraction Poles Program
(IAP 7/38 MicroMAST) initiated by the Belgian Science
Policy Office. T.G. thanks J. W. M. Bush, D. Harris, A. Oza, F.
Blanchette, R. Rosales, M. Biamonte, M. Labousse, S. Perrard,
E. Fort, Y. Couder, N. Sampara, L. Tadrist, W. Struyve, R.
Dubertrand, J.-B. Shim, M. Hubert, and P. Schlagheck for
fruitful discussions.
APPENDIX A: STABILITY OF FIXED POINTS
AND ORBITS
1. Finite memory
Fixed points of the iterated map (14) satisfy rn=r0and
wk,n =wk0. Because of axisymmetry, θncan take any constant
value, so the locus of these fixed points is a series of concentric
circles, referred to here as fixed lines. The iterated map then
becomes
wk0=μk
1−μk
ϕk0∈R,
(A1)

k
μk
1−μk
ϕk0ϕ
k0=0,
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QUANTUMLIKE STATISTICS OF DETERMINISTIC WAVE- . . . PHYSICAL REVIEW E 93, 042202 (2016)
where ϕk0=ϕk(r0) and ϕ
k0=[dϕk/dr]r0. This second condi-
tion can be written (r0)=0 with
(r)=
k
μk
1−μk
ϕ2
k(r)
2.(A2)
The stability of these solutions can be inferred from a
linearized version of the map for small perturbations: rn=
r0+˜
rn,θn=˜
θn,wkn =wk0+˜wkn, where ˜
xn1, ˜
θn1,
and ˜wkn 1. We also decompose ˜wkn =˜
ukn +i˜vkn .The
linearized map is then
˜
rn+1=˜
rn−C
k
[ϕ
k0wk0˜
rn+ϕ
k0˜
ukn],
˜
θn+1=˜
θn+C
r2
0
k
kϕk0[kwk0˜
θn+˜vkn ],
(A3)
˜
uk,n+1=μk[˜
ukn +ϕ
k0˜
rn],
˜vk,n+1=μk[˜vkn −kϕk0˜
θn].
Perturbations (˜
r,˜
uk) are decoupled from ( ˜
θ,˜vk) and can be
analyzed independently.
Radial perturbations ˜
rn=˜
r0znand ˜
ukn =˜
u0znmust satisfy
˜
uk0=μkϕ
k0/(z−μk)˜
r0as well as
1−z=C
k
μkϕk0ϕ
k0
1−μk+ϕ2
k0
z−μk.(A4)
When C1, all the solutions zshould be in the neighborhood
of z=1. If z=1−, then
=C
1−Ck
μk
(1−μk)2ϕ2
k0
(r0),(A5)
where
(r0)=
k
μk
1−μkϕk0ϕ
k0+ϕ2
k0.(A6)
Therefore, in the limit of small Cand finite damping factors
μk<1, radially stable (unstable) fixed points are found where
(r) is minimum (maximum).
Azimuthal perturbations ˜
θn=˜
θ0znand ˜vkn =˜vk0znmust
satisfy ˜vk0=−μkkϕk0/(z−μk)˜
θ0as well as
(z−1)1−C
r2
0
k
k2ϕ2
k0μk
(1−μk)(z−μk)=0.(A7)
Since z=1 is always a solution, these azimuthal perturbations
are never more than marginally stable. The other solution z
increases from zero as damping factors μkare increased (i.e.,
as the forcing amplitude is increased). Therefore, for each
fixed point r0, there is a finite threshold in forcing amplitude
for which the damping factors μksatisfy

k
k2ϕ2
k0μk
(1−μk)2=r2
0
C.(A8)
Above this threshold, there is at least one solution zlarger than
unity, so the corresponding fixed point becomes azimuthally
unstable. This corresponds to the walking threshold.
This azimuthal destabilization gives rise to periodic solu-
tions of the map (14) where the particle orbits at constant
speed around the center of the cavity: rn=r1,θn=αn and
wk,n =wk1e−ikαn (the wave pattern also rotates). The walking
velocity is then r1α. Plugging this solution in the iterated map
yields
wk1=μkϕk1
e−ikα −μk
,
r1(1−cos α)=C
k
ϕk1ϕ
k1μk
cos (kα)−μk
1−2μkcos (kα)+μ2
k
,
r2
1sin α=C
k
kϕ2
k1μk
sin (kα)
1−2μkcos (kα)+μ2
k
,(A9)
where ϕk1=ϕk(r1) and ϕ
k1=[dϕk/dr]r1. This system of
equations for (r1,α,wk1) can be solved numerically.
Slightly above the azimuthal destabilization threshold, the
orbital radius r1can be assumed to be as close to the fixed point
radius r0,sor1=r0+ε, with ε1. The angular velocity α
then satisfies
r0+C
k
μk(1+μk)
(1−μk)3k2ϕk0ϕ
k0α2
2=C(r0)ε. (A10)
Since ε→0 when α→0, orbital solutions do originate from
the azimuthal destabilization of fixed points, here through a
pitchfork bifurcation. Each orbit then directly inherits from
the radial stability of its corresponding fixed point.
Orbit stability can be inferred from a perturbation anal-
ysis rn=r1+˜
rn,θn=nα +˜
θn,wk,n =(wk1+˜wk,n)e−ikαn,
where ˜
rn1, ˜
θn1, ˜wk,n 1, and wk1=uk1+ivk1.The
linearized map is then
˜
rn+1cos α−r1sin α(˜
θn+1−˜
θn)=˜
rn−C
k
[ϕ
k1uk1˜
rn−kϕ
k1vk1˜
θn+ϕ
k1˜
uk,n],
˜
rn+1sin α+r1cos α(˜
θn+1−˜
θn)=C
r1
k
kϕ
k1vk1−ϕk1vk1
r1˜
rn+kϕk1uk1˜
θn+ϕk1˜vk,n,
˜
un+1,k =μk[cos (kα)(˜
uk,n +ϕ
k1˜
rn)−sin (kα)(˜vk,n −kϕk1˜
θn)],
˜vn+1,k =μk[cos (kα)(˜vk,n −kϕk1˜
θn)+sin (kα)(˜
uk,n +ϕ
k1˜
rn)].
Perturbations along different directions are now coupled, and eigenvalues can be found numerically.
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2. Fixed points and orbits in the case of infinite memory
Fixed points do not exist anymore at infinite memory, at
least as soon as more than one mode is selected. Indeed,
they would require ϕk(r0)=0 simultaneously for all selected
modes k. We can then look for circular orbits rn=r1,θn=αn,
wkn =wk1e−ikαn. The wave equation imposes
wk1=ϕk1
e−ikα −1,(A11)
which is finite provided that k>0 (i.e., that no purely radial
mode is selected). Then
uk1=−ϕk1
2,v
k1=ϕk1
2
sin (kα)
1−cos (kα),
r1(1−cos α)=−C
2
k
ϕk1ϕ
k1,(A12)
r2
1sin α=C
2
k
kϕ2
k1
sin (kα)
1−cos (kα).
In the limit (kα)21,
uk1=−ϕk1
2,v
k1=ϕk1
kα ,
(A13)
21+
1r1=0,r
2
1α2=C1,
with
(r)=
k
ϕk(r)2,
(r1)=1,d
dr r=r1=
1.
When the mode selection includes is at least one radial mode
(k=0), the orbit solution here above is not valid anymore,
because vk1blows up when k=0. In the iterated map, the
radial wave mode satisfies
w0,n+1=w0,n +ϕ0(rn),(A14)
so the only way to avoid having this component blow up is
to impose ϕ0(r1)=ϕ01 =0, which selects the orbital radii but
leaves u01 and v01 undetermined. Moreover, if several radial
modes coexist, there should not be any orbital solution.
The other wave components (k>0) still satisfy
uk1=−ϕk1
2,v
k1=ϕk1
2
sin (kα)
1−cos (kα).(A15)
Then the particle equations become
r1(1−cos α)=Cϕ
01u01 +2
k>0
ϕ
k1uk1,
r2
1sin α=2C
k>0
kϕk1vk1.(A16)
Again, these equations can be solved numerically to find the
orbital radii r1and their corresponding α.
APPENDIX B: WAVE ENERGY
In a recent paper, Borghesi et al. [40] observed an
equivalence between the kinetic energy of the walker and the
energy stored in the wave field, although the former was one
order of magnitude smaller than the latter. The dimensional
ballistic speed of the walker is here given by vw0.6λFf/Nc,
from which we infer the kinetic energy of the walker:
T=2π
3R3
dv2
w0.75χρ R3
dδ
R2
c
f2,(B1)
where Rdis the droplet radius. The time-averaged energy of a
mono-frequency wave field is given by
Ewave πρf 2λSH2
ndS =2π2ρf 2λ2
R2
c1
0h2
nrdr,
(B2)
where we check numerically for each subset Sthat
2π1
0h2
nrdr =
k|wk|2χ2
F
8C.(B3)
Therefore, the ratio between both energies is
T
Ewave 6
π
R3
dδ2
λ3
FR2
c
.(B4)
In this model, δis not directly expressed as a function of
other parameters, so it is unfortunately hard to conclude here
if both energies are equivalent. The estimation of δfrom more
advanced bouncing and walking models [33] is left to future
work.
[1] J.W.M.Bush,Annu. Rev. Fluid Mech. 47,269 (2015).
[2] J.W.M.Bush,Phys. Today 68(8), 47 (2015).
[3] M. Faraday, Philos. Trans. R. Soc. London 121, 319 (1831).
[4] T. Benjamin and F. Ursell, Proc. R. Soc. London, Ser. A 225,
505 (1954).
[5] K. Kumar and L. Tuckerman, J. Fluid Mech. 279,49 (1994).
[6] A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi, and Y. Couder,
J. Fluid Mech. 674,433 (2011).
[7] O. Jayaratne and B. Mason, Proc. R. Soc. London, Ser. A 280,
545 (1964).
[8] T. Gilet and J. W. M. Bush, J. Fluid Mech. 625,167 (2009).
[9] T. Gilet and J. W. M. Bush, Phys. Fluids 24,122103 (2012).
[10] Y. Couder, E. Fort, C.-H. Gautier, and A. Boudaoud, Phys. Rev.
Lett. 94,177801 (2005).
[11] T. Gilet, D. Terwagne, N. Vandewalle, and S. Dorbolo, Phys.
Rev. Lett. 100,167802 (2008).
[12] T. Gilet and J. W. M. Bush, Phys.Rev.Lett.102,014501 (2009).
[13] J. Molacek and J. W. M. Bush, J. Fluid Mech. 727,582 (2013).
[14] M. Hubert, D. Robert, H. Caps, S. Dorbolo, and N. Vandewalle,
Phys.Rev.E91,023017 (2015).
[15] Y. Couder, S. Protiere, E. Fort, and A. Boudaoud, Nature
(London) 437,208 (2005).
[16] A. Eddi, E. Fort, F. Moisy, and Y. Couder, Phys.Rev.Lett.102,
240401 (2009).
042202-14

QUANTUMLIKE STATISTICS OF DETERMINISTIC WAVE- . . . PHYSICAL REVIEW E 93, 042202 (2016)
[17] Y. Couder and E. Fort, Phys. Rev. Lett. 97,154101
(2006).
[18] C. Richardson, P. Schlagheck, J. Martin, N. Vandewalle, and
T. Bastin, arXiv:1410.1373.
[19] A. Andersen, J. Madsen, C. Reichelt, S. Rosenlund Ahl,
B. Lautrup, C. Ellegaard, M. T. Levinsen, and T. Bohr, Phys.
Rev. E 92,013006 (2015).
[20] S. Perrard, M. Labousse, M. Miskin, E. Fort, and Y. Couder,
Nat. Commun. 5,3219 (2014).
[21] E. Fort, A. Eddi, A. Boudaoud, J. Moukhtar, and Coude, Proc.
Natl. Acad. Sci. USA 107,17515 (2010).
[22] D. M. Harris and J. W. M. Bush, J. Fluid Mech. 739,444 (2014).
[23] A. U. Oza, D. M. Harris, R. R. Rosales, and J. W. M. Bush, J.
Fluid Mech. 744,404 (2014).
[24] D. M. Harris, J. Moukhtar, E. Fort, Y. Couder, and J. W. M.
Bush, Phys. Rev. E 88,011001(R) (2013).
[25] D. Shirokoff, Chaos 23,013115 (2013).
[26] B. Filoux, M. Hubert, and N. Vandewalle, Phys.Rev.E92,
041004(R) (2015).
[27] S. Perrard, M. Labousse, E. Fort, and Y. Couder, Phys. Rev. Lett.
113,104101 (2014).
[28] M. Labousse, S. Perrard, Y. Couder, and E. Fort, New J. Phys.
16,113027 (2014).
[29] A. U. Oza, R. R. Rosales, and J. W. M. Bush, J. Fluid Mech.
737,552 (2013).
[30] T. Gilet, Phys. Rev. E 90,052917 (2014).
[31] P. A. Milewski, C. A. Galeano-Rios, A. Nachbin, and J. W. M.
Bush, J. Fluid Mech. 778,361 (2015).
[32] J. W. M. Bush, A. U. Oza, and J. Molacek, J. Fluid Mech. 755,
R7 (2014).
[33] J. Molacek and J. W. M. Bush, J. Fluid Mech. 727,612 (2013).
[34] W. Batson, F. Zoueshtiagh, and R. Narayanan, J. Fluid Mech.
729,496 (2013).
[35] F. Blanchette, Phys. Fluids 28,032104 (2016).
[36] A. Eddi, D. Terwagne, E. Fort, and Y. Couder, Europhys. Lett.
82,44001 (2008).
[37] D. M. Harris and J. W. M. Bush, J. Sound Vib. 334,255 (2015).
[38] L. de Broglie, Annales de la Fondation Louis de Broglie 12,1
(1987).
[39] L. de Broglie, On the Theory of Quanta, Annales de Physique,
10e serie, t.III (1925).
[40] C. Borghesi, J. Moukhtar, M. Labousse, A. Eddi, E. Fort, and
Y. Couder, Phys. Rev. E 90,063017 (2014).
[41] S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin,
L. K. Shalm, and A. M. Steinberg, Science 332,1170 (2011).
[42] E. Nelson, Phys. Rev. 150,1079 (1966).
042202-15