Quantum-like statistics of deterministic -particle interactions in a circular cavity

Tristan Gilet1, ∗ 1Microfluidics Lab, Department of Aerospace and Mechanics, University of Liège, B-4000 Liège, Belgium A deterministic low-dimensional iterated map is proposed here to describe the interaction between a bouncing droplet and 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 Schrödinger equation for a quantum particle in a similar potential well.

PACS numbers: 05.45.Ac, 47.55.D-

I. INTRODUCTION

The interaction between Faraday waves and millimeter-sized bouncing droplets (Fig.1) has at- tracted much attention 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 vi- bration 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 λF is approximately given by the dispersion relation of water waves. The memory M is 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 sus- tained by vertically vibrating the liquid interface [10–14]. FIG. 1. (top) A drop of silicone oil bounces periodically at When the rebound dynamics locks into a periodic state the surface of a small pool (diameter 1.5 mm) of the same liquid which is vertically vibrated at 80 Hz. (bottom) Drops of with one impact every two forcing periods, the droplet appropriate size (here, diameter of 0.74 mm) can generate and becomes a synchronous emitter of Faraday waves [15]. interact with underlying Faraday waves. This wave-particle More exactly, the droplet creates a radially propagat- association is called a walker. The bottom left (right) picture ing circular capillary wave that excites standing Faraday shows a walker at low (resp. high) memory, i.e., far from waves in its wake [6]. The resulting wave field then con- (close to) the Faraday-instability threshold. tains contributions from the last M droplet impacts. In the walking state, horizontal momentum is transferred from the wave field to the impacting droplet, propor- tionally to the local wave slope at the droplet position. This coupled wave-particle entity at the millimeter scale The waves only exist in response to droplet impacts, and is called a walker. in turn they cause the horizontal motion of the droplet. The dynamics of individual walkers has been investi- gated experimentally in several configurations where it has shown properties reminiscent of quantum particles [1]. For example, walkers tunnel through weak bound- ∗ [email protected] aries [16]. They diffract or interfere when individually 2 passing through one or two slits respectively [17] (al- ents are actually necessary and sufficient for these quan- though these experimental results have recently been tum behaviors to appear. Also, no direct connection challenged by theoretical arguments [18, 19]). Walkers between the equations of motion of the walker and the also experience quantized orbits in response to confine- Schrödinger equation has been made yet. What would ment by either central forces [20], Coriolis forces [21–23] be the equivalent of Planck’s constant for walkers? On or geometry [24–26]. In the latter case, the walker evolves which timescale should the walker dynamics be averaged in a cavity of finite horizontal extent. Chaotic trajecto- to recover quantum-like statistics? This works aims at ries are often observed in the high-memory limit, when answering these questions through a theoretical investi- confinement compels the walker to cross its own path gation of walker dynamics under confinement in a two- again after less than M impacts. In response to central dimensional cavity [24]. and Coriolis forces, the walker oscillates intermittently First the model developed in Ref.[30] is particularized between several different quantized orbits, as if it were to a walker in a circular cavity (Sec. II). In Sec. III the in a superposition of trajectory eigenstates [22, 27, 28]. walker dynamics is analyzed as a function of its mem- The spatial extent of these eigenstates is always close to ory, and results are compared to the experiments of Har- an integer multiple of half the Faraday wavelength λF /2, ris et al. [24]. In Sec. IV it is shown that the present which can then be seen as the analog of the de Broglie model gives theoretical access to the experimentally- wavelength for quantum particles [21]. The probability unachievable limit of zero damping (infinite memory) and to find the walker in a given state is proportional to the perfect mode selection. Finally, the above questions are relative amount of time spent in this state. In cavity addressed through direct comparison with analytical pre- experiments [24], the probability to find the walker at a dictions from (Sec.V). given position is strongly shaped by the cavity geometry, as would be the statistical behavior of a quantum particle in a potential well. II. WAVE-PARTICLE COUPLING IN A Several theoretical models have already been pro- CONFINED GEOMETRY posed that capture various aspects of walker dynamics [6, 25, 28–31]. Most of them consider a stroboscopic point The generic model [30] of the interaction between a of view, where the droplet is assumed to impact the bath particle and a stationary wave confined to a domain S is perfectly periodically. The wave field is then expressed first recalled, here in a dimensional form. It is primarily as a sum of contributions from each successive impact [6]. based on the decomposition of the standing Faraday wave The walker horizontal trajectory is deduced from New- field Hn(X) resulting from the n first impacts into the ton’s second law, where the driving force is proportional discrete basis of eigenmodes Φk(X) of the domain: to the local wave slope and the effective mass includes X Z both the real droplet mass and an added mass from the H (X) = W Φ (X),W = H (X)Φ∗(X)dS, wave [32]. The resulting discrete iterated map can be n k,n k k,n n k k S turned into an integrodifferential equation for the walker (1) trajectory in the limit of small horizontal displacements ∗ where Φk is the conjugate of Φk. These eigenmodes are between successive impacts [29]. In an infinite space, orthonormal on S. They are here chosen to satisfy the the contribution from one droplet impact to the wave Neumann condition n·∇Φk = 0 at the boundary ∂S with field is assumed to be a Bessel function of the first kind normal vector n. J0[kF (x − xd)] centered on the droplet position xd, and We consider the stroboscopic approach where the of wavenumber kF = 2π/λF [6]. The recent model of droplet impacts the liquid bath at regular time intervals. Milewski et al. [31] provides a more sophisticated model At rebound n, it impacts at position Xn and creates a of the wave fields through inclusion of weak viscous ef- crater of shape Z = F (R) in the wave-field. In the limit fects. It also includes realistic models of vertical bounc- of a droplet size much smaller than λF , the crater F (R) ing dynamics [13, 33]. At the other end of the scale of can be approximated by a delta function weighted by the complexity, a generic model [30] has been proposed that volume of liquid displaced Ω [30]: reproduces some key features of confined wave-particle coupling and walker dynamics within a minimal math- X F (R) = Ω Φ∗(X )Φ (R) = Ωδ(R − X ). (2) ematical framework. These features include the time k n k n decomposition of the chaotic trajectory into eigenstates k [28], and the particle statistics being shaped by confine- ∗ The contribution ΩΦk(Xn) of the impact to each wave ment [24]. eigenmode depends on the droplet position. These hydrodynamic experiments and models have al- Each mode Φk(X) is given a viscous damping factor ready revealed a strong analogy between the statistical µk ∈ [0, 1] that depends on the forcing amplitude. It is behavior of chaotic walkers and quantum particles, in defined as the amplitude of mode k right before impact many different configurations. It certainly results from n + 1, divided by its amplitude right after impact n. The the deterministic chaos inherent to such wave-particle associated memory Mk is given by Mk = −1/ ln µk. The coupling. Nevertheless, it is still unclear which ingredi- Faraday threshold instability corresponds to max(µk) = 3

1. The amplitude of each mode Wk,n then satisfies the with recurrence relation Z 2π Z 1 Rc ∗ ∗ wk`,n = Wk`,n = dθ hn(r, θ)φk`rdr. (9) Wk,n+1 = µk [ΩΦk(Xn) + Wk,n] (3) Ω 0 0 n X n+1−n0 ∗ The condition w = w∗ results from h (r, θ) being = Ω µk Φk(Xn0 ). −k`,n k`,n n n0=0 real-valued. Therefore, At each impact, the particle is shifted proportionally to +∞ ∞ X X the gradient of the wave field at the impact position: hn(r, θ) = ξk< [wk`,nφk`(r, θ)] , (10) X k=0 `=1 Xn+1 − Xn = −δ Wk,n ∇Φk] (4) Xn where ξ = 2 when k > 0 and ξ = 1 when k = 0. For k k k n−1 the sake of readability, we will remove the factor ξk and X X n−n0 ∗ index ` from most notations, keeping in mind that when = −Ωδ µ Φk] ∇Φk] , Xn0 Xn cylindrical harmonics are involved, functions that depend k n0=0 on k also depend on `. Specifically, the abbreviated sum where δ > 0 is the proportionality constant between the over k will always refer to a double sum over k and `, from local wave slope and the particle displacement. Equa- 0 to ∞ and from 1 to ∞ respectively, with inclusion of tions (3) and (4) form an iterated map that describes the the factor ξk. evolution of the particle in a cavity of arbitrary shape The particle position is expressed in cylindrical coor- and dimension. dinates Xn/Rc = (rn cos θn, rn sin θn). The wave recur- rence relation (3) then becomes

 −ikθn  A. The circular cavity wk,n+1 = µk wk,n + ϕk(rn)e . (11) The trajectory equation (4) can be projected along X , Focus is now made on a two-dimensional circular cavity n then in the direction perpendicular to Xn: of radius Rc, as studied experimentally by Harris et al. [24]. Cavity eigenfunctions are ∂h  r cos(θ − θ ) = r − C n n+1 n+1 n n ∂r φ (rn,θn) k` ikθ   Φk` = , φk`(r, θ) = ϕk`(r)e , k ∈ Z, ` ∈ N0 (5) C ∂hn Rc r sin(θ − θ ) = − , (12) n+1 n+1 n r ∂θ n (rn,θn) where r = R/Rc is the dimensionless radial position, θ is the angular position, and φk` are the dimensionless where eigenfunctions. The dimensionless Faraday wavelength δΩ is defined as Λ = λ /R . The radial functions ϕ are C = (13) F F c k` R4 expressed as c is a dimensionless constant that characterizes the inten- 1 J (z r) ϕ (r) = k k` , (6) sity of the wave-particle coupling. After calculating the k` r J (z )  k2  k k` gradient of hn at the impact point, the iterated map be- π 1 − z2 k` comes

X 0  ikθn  where zk` is the `-th zero of the derivative of the Bessel rn+1 cos(θn+1 − θn) = rn − C ϕk(rn)< wk,ne 0 k function (1st kind) of order k, so Jk(zk`) = 0. The in- C teger k can take both negative and positive values. The X  ikθn  rn+1 sin(θn+1 − θn) = kϕk(rn)= wk,ne eigenfunctions satisfy orthonormality conditions rn k Z 2π Z 1 w = µ w + ϕ (r )e−ikθn  , (14) ∗ k,n+1 k k,n k n dθ φk`φk0`0 rdr = δkk0 δ``0 (7) 0 0 where ϕ0(r) = dϕ/dr. as well as the Neumann boundary condition ∂rφk` = 0 in r = 1. Some of these eigenmodes are illustrated in B. Model discussion table I. The dimensionless Faraday wave field hn(r, θ) = Hn · R2/Ω can be decomposed as a Dini series in this basis: This model of walkers neglects traveling capillary c waves, similarly to most previous works [6]. Indeed, these +∞ ∞ waves are not reenergized by the vertical forcing, and X X hn(r, θ) = wk`,nφk`(r, θ), (8) their initial energy spreads in two dimensions. Their am- k=−∞ `=1 plitude is then smaller than Faraday waves, and their 4 contribution is likely to have only marginal influence on the walker’s long-term behavior. Only when the walker TABLE I. Twelve dominant Neumann eigenmodes for a cavity of radius 14.3 mm filled with 20 cS oil and forced at 83 Hz. comes close to the boundary could these capillary waves significantly modify the local trajectory. But this effect Mode k, ` Λ µ M Mode k, ` Λ µ M would then be localized in space and time, so we assume F F that it does not strongly affect the walker statistics. 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 In contrast to previous models [29, 31], the trajectory equation (4) is here first order in time. It therefore as- sumes that the walker completely forgets its past veloc- 4,5 0.34 0.993 141 10,3 0.32 0.968 32 ity at each impact, i.e., horizontal momentum is entirely dissipated. Nevertheless, an inertial effect is still present, 13,2 0.33 0.990 99 12,2 0.34 0.962 26 because the walker keeps a constant velocity between suc- cessive impacts. An additional inertial term, e.g., corre- 17,1 0.33 0.989 90 6,4 0.34 0.951 20 sponding to the hydrodynamic boost factor [32], would increase the model complexity and the number of param- eters, so it is left to future work. 7,4 0.32 0.987 77 5,5 0.32 0.933 15

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 Φk and its associated damping factor µk. Unfortunately, these III. FINITE MEMORY IN A DAMPED WORLD modes are not easily characterized experimentally, since they cannot be excited one at a time. Uncertainty in the boundary conditions further complicates mode identifi- cation. Ideal Dirichlet boundary conditions (zero veloc- Harris and co-workers [24] considered a circular cavity of radius Rc = 14.3 mm filled with silicone oil of density ity where the liquid-air interface meets the solid walls) 3 could have been expected if the contact line was pinned 965 kg/m , surface tension σ = 20 mN/m and kinematic [34]. But in experiments contact lines are avoided on viscosity ν = 20 cS. The forcing frequency f = 70 Hz purpose because the vibration of the associated menis- was chosen in relation with the cavity size, such that the cus would be a source of parasitic capillary waves. In- most unstable mode at Faraday threshold is one of the ra- stead, the solid obstacles that provide confinement are dial modes (k = 0, ` 6= 0). In this work, damping factors always slightly submerged [16, 17, 24, 26]. The walker are approximately determined from a spectral method [5] cannot penetrate these shallow regions since waves are for this same cavity with Neumann boundary conditions strongly dissipated there [16]. Recent numerical simu- and various forcing frequencies. A frequency of 83Hz is lations suggest that such boundary condition cannot be then selected that again gives predominance to one of strictly expressed as a Robin condition [35]. From a prac- the purely radial modes. Only modes of damping factor tical point of view, when Dirichlet boundary conditions µk > 0.01 are retained. The twelve modes of highest are considered in Eq.(14), the particle tends to leave the damping factor are illustrated in Table I. Their wave- cavity as its radial displacement does not vanish at the length is always very close to the Faraday wavelength boundary. For this reason, Neumann boundary condi- that would result from this forcing frequency in the ab- tions (zero wave slope) are adopted in this work. sence of confinement.

The average walking speed vw increases with the cou- Determining the damping factor associated with each pling constant C. In experiments [24], vw = 8.66 mm/s mode is also an issue. In the absence of horizontal con- at a forcing frequency of f = 70 Hz and at 99% of the finement, viscous wave damping can be estimated by Faraday threshold. This corresponds to an average step spectral methods [5, 31]. The theoretical prediction is size 2vw/f = 0.017Rc. The same value is obtained when then validated by measuring the Faraday threshold am- Eq.(14) is solved with C = 3×10−5. If we further assume plitude as a function of forcing frequency. Unfortunately, that δ ∼ 2vw/f, the volume of fluid displaced at each confinement with submerged vertical walls yields addi- impact can be estimated from Eq.(13) to be Ω ∼ 5 µL. tional viscous dissipation. Indeed, modes φk` of high This volume corresponds to an interface deflection of a k and small ` are never observed in experiments right few hundred micrometers on a horizontal length scale of above the Faraday threshold (D. Harris, private commu- a few millimeters. It is one order of magnitude higher nication), while spectral methods would predict them as than the droplet volume, even when corrected with the highly unstable. boost factor [32]. 5

and rationalized for unconfined walkers [6, 29]. These orbits emerge from the loss of stability of their corre- sponding fixed point. The radial stability of both fixed points and corresponding orbits is related to %00(r), as al- ready shown in Ref.[30] for the unimodal and unidimen- sional version of Eqs. (3) and (4). They are usually stable 00 when % (r1) > 0 (orbits of type A) and unstable when 00 % (r1) < 0 (orbits of type B). For single frequency forc- ing, successive orbits of a same type are approximately separated by ΛF /2. The convergence of trajectories to- wards stable orbits of type A involves wobbling, i.e., ra- dial oscillations (Fig. 2).

B. Transition to chaos

100 50 30 20 15 FIG. 2. Convergence to a stable circular orbit of type A (r1 = 0.592, α = 0.0135) at memory M = 7.7 and C = 3 × 10−5. 10

The underlying wave-field corresponds to the time at which M the walker is at the position indicated by the black circle (•). 9 8 7

A. Circular orbits 6

The iterated map (14) admits periodic solutions 5 (Fig. 2) where the particle orbits at constant speed 0 0.2 0.4 0.6 0.8 1 around the center of the cavity: rn = r1, θn = αn and r −ikαn wk,n = wk1e (the wave pattern also rotates). The walking velocity is r1α. Plugging this solution into the FIG. 3. Bifurcation diagram of the radial position r as a func- iterated map yields tion of memory M at C = 3 × 10−5. The probability distri- bution function is represented in levels of gray; darker regions µk wk1 = −ikα ϕk1 (15) correspond to more frequently visited radial positions. Thick e − µk solid lines and thin dashed lines correspond to stable and un- X cos(kα) − µk stable fixed points respectively (Eq. 16). Full and empty sym- r (1 − cos α) = C ϕ ϕ0 µ 1 k1 k1 k 2 bols represent stable and unstable circular orbits respectively 1 − 2µk cos(kα) + µk k (Eq. 15). Circles (◦) indicate that at least one pair of eigen- X sin(kα) values is complex-conjugated (type A) while triangles (4) are r2 sin α = C kϕ2 µ , 1 k1 k 2 used when all corresponding eigenvalues are real-valued (type 1 − 2µk cos(kα) + µk k B). where ϕ = ϕ (r ) and ϕ0 = [dϕ /dr] . A de- k1 k 1 k1 k (r1) tailed analysis of these periodic solutions is given in Ap- pendix A. At low memory, the only solution to Eq.(15) is As memory increases, each circular orbit of type A α = 0 (fixed points). The corresponding radii r1 satisfy destabilizes radially through a Neimark-Sacker bifurca- tion (Fig. 3) [30]. Orbits of larger radius destabilize 2 0 X µk ϕk(r) at higher memory, since it takes more rebounds before % (r1) = 0, with %(r) = . (16) 1 − µk 2 the particle revisits its past positions. Sustained wob- k bling orbits of finite amplitude are then observed, where As memory increases (i.e., as all µk increase), each of the radial position oscillates periodically. As memory these fixed points experiences a pitchfork bifurcation is increased further, wobbling orbits progressively disap- where two other solutions ±α 6= 0 appear, that corre- pear. At high memory (M > 50 in Fig. 3), there are spond to clockwise and counterclockwise orbits. This bi- no stable periodic attractors left and the walker chaot- furcation is analogous to the walking threshold observed ically explores the entire cavity. The chaotic nature of 6 the system is highlighted by the exponential growth of 1 the distance between two trajectories that were initially extremely close to one other (positive Lyapunov expo- 0.8 nent in Fig. 4). 0.6

0 10 0.4

10-2 0.2

y 0 -4 10 -0.2 -0.2 | -0.4 2 -6 -r

1 10

|r -0.6 -0.4 y -0.8 10-8 -0.6 -1 -0.8 -10 -1.2 10 -0.5 0 0.5 x -1 10-12 -1 -0.5 0 0.5 1 1000 1200 1400 1600 1800 2000 x N

FIG. 5. Chaotic trajectory at memory M = 500 and C = FIG. 4. Exponential divergence of the radial distance |r −r | 1 2 3 × 10−5. The solid line corresponds to 10000 impacts, while between two neighboring trajectories, initially separated by −10 the dots emphasize 380 of these successive impacts. |r1 − r2| = 10 and placed on the same initial wave-field. Memory and coupling constants are M = 500 and C = 3 × 10−5 respectively. (Inset) The two trajectories are represented with a solid line and dots respectively, for N ∈ [1400, 1600]. finite (µk = 1, k ∈ S) while all other modes are instan- taneously damped (zero memory, µk = 0, k∈ / S). In the A chaotic trajectory is represented in Figure 5. Sev- following, several subsets of selected modes S are con- eral trajectory patterns are recurrent, including abrupt sidered (Table II). Each subset includes all the modes changes in direction and off-centered loops of character- for which the characteristic wavelength lies in a narrow istic radius close to half the Faraday wavelength ΛF /2. range λF · [1 − ∆λF , 1 + ∆λF ], except the latest subset Overall, the walker is observed to spend significantly less which is a combination of two wavelengths. Such mode time at radial positions where orbits were stable at low selection could be analog to preparing quantum particles memory (Fig. 3). with localized momentum. For subset S2 the coupling constant C is varied over four orders of magnitude.

IV. THE ACCESSIBLE INFINITE MEMORY

A. Radial statistics Unlike its hydrodynamic analog, a quantum particle confined in a cavity is known to be a Hamiltonian sys- tem for which there is no dissipation. The behavior of Figure6 shows a superposition of half a million impact individual walkers that are reminiscent of quantum par- positions. Dark circular strips indicate more frequent im- ticles were always observed at high memory, i.e., when pacts at certain radial positions r, as observed in exper- dissipation is almost balanced by external forcing. In- iments [24]. The corresponding probability distribution deed, in these conditions, the walker gets a chance to function ρ(r) shows a series of extrema at radii that are walk onto the wave field left in its own wake [20, 22, 36]. largely independent of C (Fig. 7a). Chaotic trajectories For each mode k, this balance is theoretically achieved in the infinite memory limit are qualitatively identical to when µk = 1 (infinite memory). Memory above 100 is those arising at finite memory. The radial position oscil- very challenging to achieve experimentally, owing to the lates intermittently between several preferred radii that imperfect control of the forcing vibration [37]. More- correspond to the maxima of ρ(r) [30] (Fig. 7b). over, in most geometries (including circular) cavities are The iterated map is checked to be ergodic to a good such that it is impossible to get several modes reaching approximation (within 1% in these simulations), so the µk = 1 for the same forcing acceleration. This model time-average of any property over a single trajectory co- does not suffer from such limitations; it is possible to incides with an ensemble average of this property over select a subset S of modes for which the memory is in- many independent trajectories at a given time. In other 7

TABLE II. Subsets of selected modes S, corresponding range of wavelength and coupling constant C, and associated sym- bols and colors in subsequent figures.

S Selected modes (k, `) ∈ S ΛF ∆ΛF log10 C I S1 {(0,5); (2,4); (6,3); 0.473 4% -5 (9,2); (13,1)} N S2 S1∪ {(3,4); (5,3); 0.473 8% -6.5 N (7,3);(10,2);(14,1)} -6 N -5.5 N -5 N -4.5 N -4 N -3.5 N -3 N -2.5 J S3 S2∪ {(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 FIG. 6. Two-dimensional probability distribution function (10,8); (13,7); (17,6); obtained by superimposing half a million impact positions is- (26,3); (30,2); (35,1)} sued from 78 independent trajectories. The subset of selected −5 • S5 {(0,31); (2,30); (11,26); 0.066 0.2% -7 modes is S2 (Table II), with C = 10 . (24,21); (27,20); (47,13); (62,8); (69,6); (77,4)}

? S6 S1∪ {(0,12); (2,11); 0.473 4% -5.3 So ρ(r) must always be orthogonal to each of the radial (7,9); (13,7); (17,6)} + 0.178 0.5% eigenmodes φ0(r). We have checked numerically that for every subset of selected modes S (TableII), the compo- nent of ρ(r) along φ0(r) is always at least two orders of magnitude smaller than the largest other component. words,

n+N Z 2π Z 1 1 X B. Coherence and diffusion lim f(rn0 , θn0 ) = ρ(r)f(r, θ)rdrdθ. N→∞ N n0=n 0 0 (17) The walker trajectory is usually smooth and regular at for any regular function f(rn, θn). It is rather counterin- the scale of a few impacts (Fig. 5). But such coherence tuitive that the wave-field does not blow up with time, is lost on longer timescales, where the trajectory oscil- since contributions from previous impacts keep adding up lates and loops chaotically. This behavior is quantified without any of them being damped out. The resulting through the average distance d(n) traveled in n steps, wave field amplitudes satisfy defined as

n−1 p 2 d(n) = h||x 0 − x 0 || i 0 . (21) X −ikθ 0 n +n n n wk,n = ϕk(rn0 )e n . (18) n0=0 and represented in Figure 8. For small n this distance in- creases linearly with n so the motion is ballistic (Fig. 8). In the long term, ergodicity yields As soon√ as d & ΛF /2, d starts increasing proportion- Z 2π Z 1 ally to n, like in normal diffusion. It then saturates −ikθ lim wk,n = n e dθ ρ(r)ϕk(r)rdr. (19) in d 2 owing to the finite size of the cavity. This n→∞ . 0 0 ballistic-to-diffusive transition was already observed for The angular integral vanishes by symmetry, except when the one-dimensional (1D) version of this model [30]. This k = 0. In this latter case, the only way to keep a finite suggests that the walker trajectory is similar to a random wave amplitude is to satisfy walk for which the elementary steps are of the order of ΛF /2. Z 1 The average number of impacts in one step should de- ρ(r)ϕ0(r)rdr = 0. (20) 0 pend on the walker velocity. It can be estimated by first 8

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6 n r 0.5

0.5 r

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000 ρ(r) n (a) (b)

FIG. 7. (a) Probability Distribution Function ρ(r) of radial position r, for mode subset S2 and 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 −5 particle’s radial position for mode subset S2 and for C = 10 .

3 101 40 N w 2.5 20 ) k,n 100 2 0 |> Re(w F k,n Λ 1 -20 -w

d / 1.5 F Λ 10-1 0.1 k,n+1 -40 d / 0 100 200 300 400 <|w 1 n 0.01 0.01 1 100 n / N 0.5 10-2 c

10-2 10-1 100 101 102 103 0 n / N c 0 0.2 0.4 0.6 0.8 1 r FIG. 8. Average distance d traveled in n successive impacts, FIG. 9. Average increment of wave amplitude at each impact for subsets S1, S4, S5 and S6. The solid line corresponds h|wk,n+1 −wk,n|i as a function of particle position r, for modes to the ballistic regime d/ΛF = an/Nc, while the dashed line corresponds to the diffusive regime d/Λ = bpn/N . (In- of the subset S2. The solid line is |ϕk(r)|. Modes (k, `) = F c (0, 5) and (6, 3) are represented in black and gray, respectively. set) Same plot, for S1, S2 and S3. Symbols and colors are explained in Table II. Inset shows an example of the time evolution of the wave amplitude <(wk,n), for these same modes. Nw is defined as the distance between two successive extrema of wk,n. looking at the amplitude wk,n of each wave eigenmode (see inset of Fig. 9). Their time evolution is a succession of coherent linear segments of variable duration N . This w N peaks at approximately the same value for each constant growth rate depends on the particle position; at w mode of a given selection, then it decreases exponen- every impact, each wk increases by an average amount tially for large Nw (Fig. 10). So after the average co- herence time hN i each mode amplitude w has grown h|wk,n+1 − wk,n|in ' |ϕk(r)| (22) w k by about hNwi|ϕk| and its contribution to the total 2 as confirmed in Fig. 9. The statistical distribution of wave slope now scales as ξkwk∇ϕk ∼ 2ξkNcϕk/ΛF . 9

0.04 0.3 where b ' 0.62 based on the data of subset S5, for which 0.035 0.2 the diffusive region is the largest since ΛF is the smallest

w (Fig. 8). If the diffusive behavior is attributed to a two- L 0.03 0.1 dimensional (2D) random walk, then the corresponding diffusion coefficient is defined as 0 0.025 0 0.1 0.2 0.3 2 2 2 Λ /2 d b Λ F ˜ F 0.02 D , = . (26) 1000 4n 4 Nc PDF 100 0.015 > The crossover between both regimes occurs in n/Nc = w b2/a2 ' 1.2, so in d/Λ = b2/a ' 0.67. It can be seen as

Substitution in Eq.(29) indeed yields cylindrical harmonics φk(r) defined in Eq.(5): 1 X −iωkt 2 2 Ψ = ckφk(r)e , (32) i ∂ Ψ + ~ ∇2Ψ = ~ ∂ Ψ. (31) Rc ~ t 2m 2mc2 tt k 2 2 2 where (Rc ∇ + zk)φk = 0 and The left hand side is identical to Eq.(28), so the right hand side is a relativistic correction. The characteris- ω = ~ z2 (33) k 2mR2 k tic frequency of Ψ(x, t) is ω = ~k2/(2m). Therefore, c the relativistic correction is negligible when ω/k  c, or in order to satisfy the Schrödinger equation. The proba- equivalently when kλc  1, where λc = 2π~/(mc) is the bility density function (PDF) is then Compton wavelength. X ∗ ∗ −i(ωk−ωj )t Couder and co-workers have already observed in many ρ(r, t) = cj ckφj (r)φk(r)e . (34) different configurations [17, 20, 21] that the Faraday j,k wavelength λF was the walker equivalent of the de Broglie R P 2 Coefficients ck must satisfy S ρdS = k |ck| = 1. Since wavelength λdB = 2π/k of quantum particles (Analogy all zk are distinct (there is no degeneracy), the time- No. 1). In a recent review paper, J.W.M. Bush [1] pushed averaged probability density is the analogy one step further and hypothesized that the X 2 X 2 2 bouncing motion of walkers could be the equivalent of ρ(r) = |ckφk(r)| = |ck| ϕk(r) , (35) this Zitterbewegung, so the bouncing frequency f would k k 2 correspond to the Compton frequency mc /(2π~) (Anal- which is necessarily axisymmetric. Although all zeros ogy No. 2). of bessel functions (and derivatives) are strictly dis- The diffusive behavior of the walker on the long tinct, it is always theoretically possible to find and se- term suggests a third analogy (No. 3), between the lect two eigenmodes (j, k) for which |zj − zk| is arbitrar- walker’s diffusion coefficient D and the coefficient of ily small. The corresponding imaginary exponential in Schrödinger equation ~/m. It is equivalent to say that Eq.(34) would generate some beating at the extremely 2 the Schrödinger timescale 2π/ω = 4πm/(~k ) is anal- low frequency (ωj − ωk) that would challenge any practi- ogous to the dimensional coherence time of the walker cal computation of a time average. Nevertheless, as far as 2 2 −1 −1 λF /(πD) ' [4/(πb )]Ncf ' 3.3Ncf . De Broglie’s the modes selected in this work are concerned (Table II), momentum relation states that the speed of a nonrela- |zj − zk| > 0.01 for any j 6= k. tivistic particle is v = ~k/m, which is then equivalent How well are the walker statistics described by these 2 −1 −1 to [πb /2]λF fNc ' 0.60λF fNc for the walker. This quantum predictions? Following de Broglie’s pilot wave value is remarkably close to the ballistic speed aλF f/Nc theory [38, 39], we assume an analogy between the quan- with a = 0.57 as observed in Fig. 8. The analog of the tum wave function and the classical wave field of the p −1/2 −1/2 speed of light c is [b π/2]λF fNc = 0.78λF fNc , walker, averaged over the coherent timescale. In Sec. IV, which does not seem to be a fundamental constant in the the walker dynamics was considered in configurations walker’s world. Nevertheless, the relativistic limit v → c where a small number of modes were equally excited 2 corresponds to Nc → πb /2 ' 0.60 for the walker, i.e., while others were strictly not. This suggests the iden- the coherence time becomes of the order of the bouncing tification of quantum coefficients ck as period. Saying that a quantum particle cannot go faster ξk than light is then analogous to saying that the coher- c = (36) k pP 2 ence time of a walker trajectory should be at least one k ξk 2 rebound time. When Nc = πb /2 ' 0.60, the equivalent where again ξk = 1 if k = 0 and ξk = 2 otherwise. As seen speed of light becomes λF f, which is the phase speed in Fig. 7a, the corresponding quantum prediction of the of capillary waves. Finally, in quantum mechanics the average probability density (Eq. 35) is reminiscent but correspondence principle states that classical mechanics not identical to the one obtained from walker simulations. is recovered in the limit ~ → 0. This translates into Nevertheless, their extrema of density coincide almost the walker behavior being fully ballistic when D goes to perfectly. zero, i.e., when the coherence timescale Nc goes to in- finity. The three equivalences and their implications are summarized in Table III. C. Average kinetic energy

In quantum mechanics, the kinetic energy operator 2 Tˆ = − ~ ∇2 is Hermitian, and its expected value B. Probability density 2m 2 Z 2 X hT i = hΨ|Tˆ|Ψi = − ~ Ψ∗∇2ΨdS = ~ |c |2z2 2m 2mR2 k k In a 2D circular infinite potential well of radius Rc, the S c k wave-function can be decomposed into a discrete basis of (37) 11

TABLE III. Correspondence of variables between walkers and quantum particles (b ' 0.62).

Analogy Walker Quantum particle

# 1 Faraday wavelength λF de Broglie wavelength 2π/k 2 # 2 Bouncing frequency f Zitterbewegung frequency mc /(2π~) 2 2 −1 # 3 Diffusion coefficient D = [b /4]λF fNc Schrödinger diffusion ~/m 2 −1 2 Coherence time [4/(πb )]Ncf Schrödinger timescale 4πm/(~k ) 2 −1 Ballistic speed v = [πb /2]λF fNc Quantum speed v = ~k/m Derived p −1/2 Maximum speed [b π/2]λF fNc Speed of light c 2 Relativistic limit Nc > πb /2 Relativistic limit v/c < 1 Ballistic limit Nc → ∞ Classical limit ~ → 0

0 is time-independent in a infinite potential well. A dimen- 10 sionless kinetic energy can then be defined based on the cavity radius Rc and Compton frequency ωc/(2π):

2 10-2 2 4π 2 X 2 2 Te = hT i = B |ck| z (38) m R2ω2 k c c k

2 where B = 2π /(mR ωc) is a dimensionless coefficient. walker -4 ~ c e T 10 According to Table III, B is equivalent to the dimen- sionless diffusion coefficient D˜ of the walker, defined in Eqs. (26) and (27). This quantum prediction is in re- markable agreement with the observed average kinetic 10-6 energy of simulated walkers, for all considered mode sub- sets (Fig. 11). Only simulations at large C fail to match the prediction, possibly because they would be in the rel- 10-6 10-4 10-2 100 ativistic regime (N < 5). The energy of the mixed-mode e c Tquantum subset S6 is also overestimated. The kinetic energy of two interacting walkers was shown to be somehow equiv- FIG. 11. Average dimensionless kinetic energy Te of the alent to the energy stored in the corresponding wave field walker, vs quantum prediction, with the equivalence B ≡ D˜. [40]. Nevertheless, as detailed in Appendix B, it is here Symbols correspond to different mode subsets (Table II). The unclear if both energies are still equivalent for a single solid line is the quantum prediction (Eq. 38). walker confined in a cavity.

D. Position-dependent statistics "weakly" measure a quantum particle, gaining some in- Radial- and azimuthal-velocity operators are defined formation about its momentum without appreciably dis- turbing it, so its position can be "strongly" measured by projecting the momentum operator Pˆ = −i~∇ along the radial and azimuthal directions, respectively: directly after [41]. The information obtained from in- dividual measurements is limited. But one can perform many trials, then postselect particles that were observed Vˆ = 1 e · Pˆ = −i ~ ∂ , r m r r at a given position and calculate their associated average mRc momentum. We here propose to extend this concept of ˆ 1 ˆ ~ 1 Vθ = eθ · P= −i ∂θ (39) weak measurement to a particle in a 2D circular cavity. m mR r c We define the Hermitian operators On the one hand, the radial-velocity operator is not Her- mitian, so it should not be observable. On the other hand, the tangential-velocity operator has a trivial ex- δˆ(r − R) δˆ(r − R) Vˆ 2 = Vˆ † Vˆ and Vˆ 2 = Vˆ † Vˆ . pected value of zero, by symmetry. The Heisenberg un- r r 2πR r θ θ 2πR θ certainty principle states that one cannot measure accu- (40) rately and simultaneously both the position and momen- which are aimed to represent the squared radial and az- tum of a quantum particle. However, it is possible to imuthal velocities at a given radial position R, respec- 12

200 tively. Their expected values 180  2 Z ˆ 2 ~ ∗ δ(r − R) hΨ|Vr |Ψi = ∂rΨ ∂rΨ dS, (41) 160 mRc S 2πR 140  2 Z δ(r − R) hΨ|Vˆ 2|Ψi = ~ ∂ Ψ∗∂ Ψ r−2dS θ θ θ 120 mRc S 2πR 2 ˜ D 100 / 2

correspond to the variance of each velocity component. θ e v They are time-averaged 80

 2 60 ˆ 2 ~ X 2 2 hΨ|Vr |Ψi = |ck| [∂rϕk]r=R , (42) mRc 40 k  2 20 ˆ 2 ~ X 2 2 2 hΨ|Vθ |Ψi = |ck| k [ϕk]r=R , (43) mRc 0 k 0 0.2 0.4 0.6 0.8 1 and made dimensionless R

2 2 ˆ 2 4π 2 X 2 2 2 ver = hΨ|Vr |Ψi 2 2 = B |ck| [∂rϕk] , FIG. 13. Average variance of the tangential velocity ve of Rc ωc r=R θ k the walker at a given radial position r, normalized by D˜ 2, for 2 ˆ 2 4π2 2 X 2 2 2 subset S2 (Table II). The solid line corresponds to 2.0 times ve = hΨ|V |Ψi 2 2 = B |ck| k [ϕk] . (44) θ θ Rc ωc r=R the quantum prediction [Eq. (44); best fit]. k

600 considered (Fig. 14). However, the quantum calculation 500 strongly underestimates both walker variances at large C (relativistic regime). These results demonstrate how much the velocity 400 statistics of the walker are shaped by the wave function in almost the same way as the statistics of a quantum 2 ˜ D 300 particle would be. Moreover, they lead to an interpreta- / 2 r e v tion of the Heisenberg uncertainty principle for walkers (slightly different from the one proposed in Ref.[17]). The 200 uncertainty in position can be related to the coherence length and it is then of the order of λF /2 (equivalently 100 λdB/2). The uncertainty in speed is directly given√ by the dimensional version of the standard deviation ve, which scales as D/λ [equivalently /(mλ )]. The product of 0 F ~ dB 0 0.2 0.4 0.6 0.8 1 these uncertainties then scales as the diffusion coefficient R D, which is equivalent to ~/m. So the uncertainty prin- ciple is recovered for walkers provided their dynamics is 2 FIG. 12. Average variance of the radial velocity ver of the analyzed at the scale of diffusion. walker at a given radial position r, normalized by D˜ 2, for subset S2 (Table II). The solid line corresponds to 3.16 times the quantum prediction [Eq. (44); best fit]. VI. DISCUSSION

Again, the equivalence between B and D˜ allows for The theoretical framework introduced in Sec.II is one a direct comparison between walkers and quantum pre- of the simplest mathematical representations of a parti- cle coupled to a wave. It qualitatively captures the key dictions. The evolution of v2 with R is very similar in er features observed in the experiments on confined walk- both worlds, although the proportionality coefficient be- ers performed by Harris et al. [24]. It reproduces both tween both (best fit) varies from one subset to another the circular orbits at low memory and the chaotic tra- 2 (Fig. 12). The agreement is even better for veθ , where jectories at high memory. The current limitation for the walker variance is almost exactly twice the quantum a quantitative comparison originates in the lack of ex- prediction, at any radial position for most values of C perimental information about the damping rate of each (Fig. 13). This remarkable similarity holds for any of the eigenmode. Nevertheless, this model allows for the explo- 2 markedly different functions veθ (R) at each mode subset ration of regimes that are not accessible experimentally. 13

as analogous to the preparation of a quantum state with a more-or-less defined momentum. Nevertheless, it must 150 be noted that this walker model is still highly dissipa- tive since the nonselected modes (the ones that do not resonate with the forcing) are immediately damped.

2 100 ~

D The chaotic trajectories of confined walkers are ballis-

=

2

3 e v tic on the short term and diffusive on the long term. The 50 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] 0 also confirms. The Faraday wavelength is at the heart of most quantum-like behaviors of walkers; it is identified as equivalent to the de Broglie wavelength for a quantum 1000 particle. Our analysis of the diffusive motion in a cir-

2 cular corral has suggested another equivalence, between

~ D

= the diffusion coefficient D and the factor ~/m in the

2

3 e v Schrödinger equation. The walker behavior thus becomes 500 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 0 closely to the de Broglie speed ~k/m. Similarly, the av- erage kinetic energy of the walker matches Schrödinger’s prediction over several orders of magnitude.

10000 The bouncing dynamics of the walkers was previously 2

~ hypothesized as reminiscent of the Zitterbewegung of

D

= 2

3 quantum particles [1]. From there, we found an analog

e v 5000 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 0 mathematical framework allows for a future investigation of this regime, which is unfortunately not attainable in 300 experiments where the walking steps are usually limited

to around λF /20 [6, 24]. Equivalence relations of TableIII 2

~ do not explicitly depend on the dispersion relation of the

D 200

= 2

3 waves. Nevertheless, the variables therein (such as the e v coherence time or the elementary diffusion coefficient) 100 do depend on the considered wave-particle interaction, e.g., here through the coupling constant C [Eq. (23)]. 0 This might be the reason why the analogs for the speed 0 0.2 0.4 0.6 0.8 1 of light and the Planck constant are not constant in the R walker’s world. Harris et al. [24] observed that the statistics of con- 2 fined walkers can be shaped by the cavity eigenmodes FIG. 14. Variance of the tangential velocity veθ of the walker at a given radial position r, normalized by D˜ 2, for subsets in the high-memory limit. We have here calculated S1, S4, S5, and S6, from top to bottom (Table II). The solid the position-dependent variance of walker velocity in the lines in correspond, respectively, to 1.0, 2.1, 2.2 and 2.0 times limit of some modes having an infinite memory. Their the quantum prediction associated with each subset [Eq. (44); dependence on radial position is remarkably close to the best fit]. predictions obtained from the quantum formalism (linear Schrödinger equation) with Hermitian observables, even when complex combinations of modes are considered. This model of walkers is reminiscent of the pilot-wave Of particular interest is the possibility to set the memory theories of de Broglie and Bohm, although there are some of each mode to either infinity (no damping) or zero (full significant differences [1]. In the Bohmian mechanical de- damping). Quantumlike behavior of individual walkers scription of quantum mechanics, particles are guided by a were observed at high-memory, when the system was as pilot-wave prescribed by Schrödinger’s equation. It thus close to conservative as it can be. In this model, the evolves at the Schrödinger timescale. The particles do mode selection around one given wavelength can be seen not exert any direct individual feedback on this wave; 14 only their statistics shapes the wave. By contrast, indi- Stability of these solutions can be inferred from a vidual bouncing walkers locally excite the wave field that linearized version of the map for small perturbations: ˜ sets them into motion. The double-solution theory of rn = r0 +r ˜n, θn = θn, wkn = wk0 +w ˜kn, where ˜ de Broglie [38] involves an additional pilot-wave centered x˜n  1, θn  1 and w˜kn  1. We also decompose on the particle, whose timescale would be the Zitterbewe- w˜kn =u ˜kn + iv˜kn. The linearized map is then gung period. This second wave could be the analog of the X 00 0 real Faraday wave that couples with individual walkers. r˜n+1 =r ˜n − C [ϕk0wk0r˜n + ϕk0u˜kn] It was already shown that the Schrödinger equation can k be retrieved from a random walk of diffusion coefficient C X h i θ˜ = θ˜ + kϕ kw θ˜ +v ˜ /(2m) [42]. This work suggests that such a random walk n+1 n r2 k0 k0 n kn ~ 0 k can originate from the chaos of a deterministic map that 0 describes the coupling of a wave and a particle. In other u˜k,n+1 = µk [˜ukn + ϕk0r˜n] h ˜ i words, the solution of Schrödinger equation for a particle v˜k,n+1 = µk v˜kn − kϕk0θn (A3) in a cavity can be obtained from a purely deterministic mechanism that does not involve any stochastic element. ˜ Perturbations (˜r, u˜k) are decoupled from (θ, v˜k) and can Walkers have now been investigated for a decade. They be analyzed independently. have shown many behaviors reminiscent of quantum par- n n Radial perturbations r˜n =r ˜0z and u˜kn =u ˜0z must ticles. We have shown here that, when they are confined 0 satisfy u˜k0 = µkϕ /(z − µk)˜r0 as well as in cavities, their statistics closely approaches the solution k0 of the Schrödinger equation. Future work is still required  00 02  X ϕk0ϕk0 ϕk0 to identify the exact limits of this analogy. 1 − z = C µk + (A4) 1 − µk z − µk k When C  1, all the solutions z should be in the neigh- ACKNOWLEDGMENTS borhood of z = 1. If z = 1 − , then C This research was performed in the framework of the  = %00(r ) (A5) P µk 02 0 Quandrops project, financially supported by the Ac- 1 − C 2 ϕ k (1−µk) k0 tions de Recherches Concertees (ARC) of the Federation Wallonie-Bruxelles under contract No. 12-17/02. This where research has also been funded by the Interuniversity At- 00 X µk  00 02  traction Poles Program (IAP 7/38 MicroMAST) initi- % (r0) = ϕk0ϕk0 + ϕk0 (A6) 1 − µk ated by the Belgian Science Policy Office. T.G. thanks k J.W.M. Bush, D. Harris, A. Oza, F. Blanchette, R. Ros- Therefore in the limit of small C and finite damping fac- ales, M. Biamonte, M. Labousse, S. Perrard, E. Fort, tors µk < 1, radially-stable (resp. unstable) fixed points Y. Couder, N. Sampara, L. Tadrist, W. Struyve, R. Du- are found where %(r) is minimum (resp. maximum). bertrand, J.-B. Shim, M. Hubert and P. Schlagheck for ˜ ˜ n n Azimuthal perturbations θn = θ0z and v˜kn =v ˜k0z fruitful discussions. µkkϕk0 ˜ must satisfy v˜k0 = − θ0 as well as z−µk  C k2ϕ2 µ  Appendix A: Stability of fixed points and orbits X k0 k (z − 1) 1 − 2 = 0 (A7) r (1 − µk)(z − µk) 0 k 1. Finite memory Since z = 1 is always a solution, these azimuthal per- turbations are never more than marginally stable. The Fixed points of the iterated map (14) satisfy rn = r0 other solution z increases from zero as damping factors µk and wk,n = wk0. Because of axisymmetry, θn can take are increased (i.e., as the forcing amplitude is increased). any constant value, so the locus of these fixed points is a Therefore, for each fixed point r0, there is a finite thresh- series of concentric circles, referred to here as fixed lines. old in forcing amplitude for which the damping factors The iterated map then becomes µk satisfy µk wk0 = ϕk0 ∈ R k2ϕ2 µ r2 1 − µk X k0 k 0 2 = (A8) µk (1 − µk) C X 0 k ϕk0ϕk0 = 0 (A1) 1 − µk k Above this threshold, there is at least one solution z 0 where ϕk0 = ϕk(r0) and ϕk0 = [dϕk/dr]r0 . This second larger than unity, so the corresponding fixed point be- 0 condition can be written % (r0) = 0 with comes azimuthally unstable. This corresponds to the 2 walking threshold. X µk ϕ (r) %(r) = k (A2) This azimuthal destabilization gives rise to periodic 1 − µk 2 k solutions of the map (14) where the particle orbits at 15 constant speed around the center of the cavity: rn = r1, fixed point radius r0, so r1 = r0 + ε, with ε  1. The −ikαn θn = αn and wk,n = wk1e (the wave pattern also angular velocity α then satisfies rotates). The walking velocity is then r1α. Plugging this solution in the iterated map yields " # 2 X µk(1 + µk) 2 0 α 00 µkϕk1 r0 + C 3 k ϕk0ϕk0 = C% (r0)ε w = (A9) (1 − µk) 2 k1 e−ikα − µ k k (A10) cos(kα) − µ X 0 k Since ε → 0 when α → 0, orbital solutions do originate r1(1 − cos α) = C ϕk1ϕk1µk 2 1 − 2µk cos(kα) + µ k k from the azimuthal destabilization of fixed points, here through a pitchfork bifurcation. Each orbit then directly 2 X 2 sin(kα) r1 sin α = C kϕk1µk 2 inherits from the radial stability of its corresponding fixed 1 − 2µk cos(kα) + µ k k point. where ϕ = ϕ (r ) and ϕ0 = [dϕ /dr] . This system Orbit stability can be inferred from a perturbation k1 k 1 k1 k r1 ˜ of equations for (r1, α, wk1) can be solved numerically. analysis rn = r1 +r ˜n, θn = nα + θn, wk,n = (wk1 + −ikαn ˜ Slightly above the azimuthal destabilization threshold, w˜k,n)e , where r˜n  1, θn  1, w˜k,n  1, and the orbital radius r1 can be assumed to be as close to the wk1 = uk1 + ivk1. The linearized map is then

˜ ˜ X h 00 0 ˜ 0 i r˜n+1 cos α − r1 sin α(θn+1 − θn) =r ˜n − C ϕk1uk1r˜n − kϕk1vk1θn + ϕk1u˜k,n k    ˜ ˜ C X 0 ϕk1vk1 ˜ r˜n+1 sin α + r1 cos α(θn+1 − θn) = k ϕk1vk1 − r˜n + kϕk1uk1θn + ϕk1v˜k,n r1 r1 k h 0  ˜ i u˜n+1,k = µk cos(kα) (˜uk,n + ϕk1r˜n) − sin(kα) v˜k,n − kϕk1θn h  ˜  0 i v˜n+1,k = µk cos(kα) v˜k,n − kϕk1θn + sin(kα) (˜uk,n + ϕk1r˜n) .

Perturbations along different directions are now coupled, In the limit of (kα)2  1, and eigenvalues can be found numerically. ϕ ϕ u = − k1 , v = k1 k1 2 k1 kα 2% + %0 r = 0, r2α2 = C% , (A13) 2. Fixed points and orbits in the case of infinite 1 1 1 1 1 memory with

Fixed points do not exist anymore at infinite memory, X d% %(r) = ϕ (r)2,%(r ) = % , = %0 . at least as soon as more than one mode is selected. In- k 1 1 dr 1 k r=r1 deed, they would require ϕk(r0) = 0 simultaneously for all selected modes k. We can then look for circular orbits −ikαn rn = r1, θn = αn, wkn = wk1e . The wave equation When the mode selection includes is at least one ra- imposes dial mode (k = 0), the orbit solution here above is not valid anymore, because vk1 blows up when k = 0. In the ϕ iterated map, the radial wave mode satisfies w = k1 , (A11) k1 e−ikα − 1 w0,n+1 = w0,n + ϕ0(rn) (A14) which is finite provided that k > 0 (i.e., that no purely radial mode is selected). Then so the only way to avoid having this component to blow up is to impose ϕ0(r1) = ϕ01 = 0, which selects the ϕ ϕ sin(kα) u = − k1 , v = k1 orbital radii but leaves u01 and v01 undetermined. More- k1 2 k1 2 1 − cos(kα) over, if several radial modes coexist, there should not be C X any orbital solution. r (1 − cos α) = − ϕ ϕ0 1 2 k1 k1 The other wave components (k > 0) still satisfy k C X sin(kα) r2 sin α = kϕ2 . (A12) ϕk1 ϕk1 sin(kα) 1 2 k1 1 − cos(kα) uk1 = − , vk1 = (A15) k 2 2 1 − cos(kα) 16

Then the particle equations become where Rd is the droplet radius. The time-averaged energy " # of a mono-frequency wave field is given by 0 X 0 r1(1 − cos α) = C ϕ01u01 + 2 ϕk1uk1 Z Ω2 Z 1 k>0 E ' πρf 2λ hH2idS = 2π2ρf 2λ hh2 irdr X wave n R2 n r2 sin α = 2C kϕ v (A16) S c 0 1 k1 k1 (B2) k>0 where we check numerically for each subset S that Again, these equations can be solved numerically to find 1 the orbital radii r1 and their corresponding α. Z X χΛ2 2π hh2 irdr = h|w |2i ' F . (B3) n k 8C 0 k Appendix B: Wave energy Therefore, the ratio between both energies is In a recent paper, Borghesi et al. [40] observed an 3 2 equivalence between the kinetic energy of the walker and T 6 Rdδ ' 3 2 (B4) the energy stored in the wave field, although the former Ewave π λF Rc was one order of magnitude smaller than the latter. The dimensional ballistic speed of the walker is here given by In this model, δ is not directly expressed as a function of vw ' 0.6λF f/Nc, from which we infer the kinetic energy other parameters, so it is unfortunately hard to conclude of the walker: here if both energies are equivalent or not. The esti- 3 mation of δ from more advanced bouncing and walking 2π 3 2 RdδΩ 2 T = Rdvw ' 0.75χρ 2 f (B1) models [33] is left to future work. 3 Rc

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