Non-Hamiltonian features of a classical pilot-wave dynamics

A bouncing droplet on a vibrated bath can couple to the waves it generates, so that it becomes a propagative walker. Its propulsion at constant velocity means that a balance exists between the permanent input of energy provided by the vibration and the dissipation. Here we seek a simple theoretical description of the resulting non-Hamiltonian dynamics with a walker immersed in a harmonic potential well. We demonstrate that the interaction with the recently emitted waves can be modeled by a Rayleigh-type friction. The Rayleigh oscillator has well defined attractors. The convergence toward them and their stability is investigated through an energetic approach and a linear stability analysis. These theoretical results provide a description of the dynamics in excellent agreement with the experimental data. It is thus a basic framework for further investigations of wave-particle interactions when memory effects are included.

Figure 1

Sketch of the experimental setup. (a) A bath of silicon oil is set on a vertically oscillatory motion at a frequency $f_0=80$ Hz. A drop deposited on the surface starts bouncing without coalescence (prevented by the air layer between the liquid surface and the drop). A small amount of ferrofluid is then encapsulated inside the drop so that it becomes sensitive to a magnetic field. A current $I_0$ through two coils in a Helmholtz position generates a homogeneous magnetic field at the bath surface. By adding an additional magnet at a distance $d$ above the surface, an attractive force can be generated on the drop. The resulting potential can be modeled by a harmonic well: $E_p=-\kappa r^2/2$ (red arrow online). (b) Details of the walker's motion. The walker moves at a velocity $V_0$ due to the propulsion by damped Faraday waves. These waves (indicated by large open circles in the figure) are generated by the impact of the drop (indicated by small, solid circles) and are damped with a characteristic time $\tau$ larger than the vertical period of motion. Their decay time $\tau$ can be converted in a memory length $L=\tau V_0$. Here, we study the limit in which $L$ is small compared to any curvature radius of the trajectory.

Figure 2

(a) Chronophotography of the transient motion of the walker being trapped by the magnetic field. The time increment is fixed at 0.05 s which corresponds to one point every two bounces. After a few oscillations, the walker trajectory converges to a circular orbit. (b) Transient regimes obtained with Fort's model of the walker dynamics (solid black line) and numerical solving of the Rayleigh equation with $\Gamma =25.5$ [solid red line (online), light gray (printed)]. The quantitative agreement between experiments and numerical simulations shows that the walker propulsion can be described through a friction term depending on the velocity. (c) Orbit radius $R/{\lambda }_F$ as a function of the dimensionless frequency ${\lambda }_F\omega /V$, where $\omega$ is the characteristic frequency of the drop in the harmonic well. Experiments for two drops of velocity $V=10$ mm/s and 8 mm/s ($\circ$). The experimental data (red dots online) are compared to the simple scaling $R/{\lambda }_F=V/\omega {\lambda }_F$ (solid black line downer). The agreement is good, without use of any fit parameter. The blue (upper) line (blue online, gray printed) indicates the result of Fort's numerical model.

Figure 3

Experimental characterization of the transient behavior. The magnetic force changes abruptly every ten seconds, giving rise to repeated transients to the asymptotic radius. For each time interval, the experimental radius (oscillating black solid line) and the expected asymptotic radius (solid red line online) are shown. (b) Superposition of ten experimental transients for a final imposed radius of $R_1=0.4$ mm (black thin solid lines) corresponding to an imposed $\omega ={\omega }_1$. The red thick solid line indicates the average of the experimental transients. The radius converges by oscillating to the attractor without overshoot. (c) Superposition of ten experimental transients in the case of a final imposed radius of $R_2=4.8$ mm (thin black solid lines) corresponding to an imposed $\omega ={\omega }_2$. The red thick solid line indicates the average of the experimental transients. The radius converges to the attractor through damped oscillations.

Figure 4

Dimensionless characteristic rate $T_R/T_c$ of convergence to the circular motion ($T_R$ is the orbiting period) as a function of $R/V_0$: experimental data (red circle online), numerical solving of the Rayleigh oscillator (black dashed line). A larger value means a faster convergence. Numerically, we change $\omega =1/\left(\Gamma T_V\right)$ by decreasing $\Gamma$ with $T_V=0.14\pm 0.01$ s. A minimum in the convergence time is observed both in the experiment and in the theory.

Figure 5

Stability of the circular attractor. (a) Schematics to illustrate the stability of the limit cycle $(r=r_0=1,v=v_0=\pm 1)$. The evolution of $\dot{E}$ with the square of the dimensionless speed $v^2$ (in black, right $y$ axis). The evolution of the energy $E$ with $v^2$ for a given dimensionless radius $r=1$ (in blue online, light gray printed, left $y$ axis). Two fixed points $\dot{E}=0$ are identified: $v=0$ unstable [$d\dot{E}/d\left(v^2\right)>0$] and $v=\pm 1$ stable [$d\dot{E}/d\left(v^2\right)<0$]. At the equilibrium $(r=1,v=1)$, there is an equipartition of energy between the kinetic energy $E_k^{\mathrm{eq}}$ and the potential energy $E_p^{\mathrm{eq}}$. (b) Time evolution of the total energy as a function of $v^2$. At large time, the dynamics reaches the fixed point $(E=1,v=\pm 1)$, i.e., $(r=1,v=\pm 1)$. The case of large $\Gamma$ is in the solid back line $(\Gamma =70$) while the blue dashed line indicates the case of small $\Gamma =0.1$. (c) Trajectories corresponding to the case in part (b).

Figure 6

(a) Spectrum of the position $x\left(t\right)$. The frequencies are normalized by the fundamental frequency $f_0$. Amplitude spectrum $|\widehat{x}\left(f\right)|$ of the paths obtained, from experiments [blue (upper) solid line online] from Fort's model (black solid lines) and from the numerical resolution of the Rayleigh oscillator (red solid line online, gray printed, overlapping the black curve) with $\Gamma =25.5$. In green (online, for the printed version downer line at $f/f_0=1$), we plot the spectrum of the transient of Eq. (5). In vertical dashed lines are represented the position of the predicted eigenfrequency $f_1^{\pm }=\sqrt{2}\pm 1$. (b) The phase spectrum $\Phi$ of $x\left(t\right)$ of a transient ruled by the two dimensional Rayleigh equation (in grey), and in the corresponding smoothed curve (in black). We indicate by the vertical dashed line the higher order eigenfrequencies $f_n^{+}=n+\sqrt{n+1}$ of the Rayleigh oscillator. (c),(d) Real and imaginary parts of the Floquet coefficients ordered according to their asymptotic value $\Gamma \gg 1$. Black circles ${\mu }_s\sim -2\Gamma +i$. Green (online, black printed version) points, ${\mu }_0=i$. Blue (light gray, printed version) circles and blue (light gray, printed version) points, $\mu =\sim \pm (\sqrt{2}-1)i-O(1/\Gamma )$ giving way to $f_1^{\pm }$ [31].