Simulations of pilot-wave dynamics in a simple harmonic potential

Abstract

We present the results of a numerical investigation of droplets walking in a harmonic potential on a vibrating fluid bath. The droplet's trajectory is described by an integrodifferential equation, which is simulated numerically in various parameter regimes. We produce a regime diagram that summarizes the dependence of the walker's behavior on the system parameters for a droplet of fixed size. At relatively low vibrational forcing, a number of periodic and quasiperiodic trajectories emerge. In the limit of large vibrational forcing, the walker's trajectory becomes chaotic, but the resulting trajectories can be decomposed into portions of unstable quasiperiodic states.

Hydrodynamic spin states

“…In this regime, the walker's inertia is negligible and the trajectory equation (6) is firstorder in time. This parameter regime thus might be ideal for exploring other hydrodynamic quantum analogs, such as the double quantization of trajectories in radial extent and angular momentum 4,29,30 emerging in the presence of applied potentials. The discrete-time theoretical model of Durey and Milewski 29 represents an efficient means to address this class of problems.…”

Walking droplets in a circular corral: Quantisation and chaos

“…We note that these two time scales are comparable, indicating that we are in the memory-dominated regime in which the entirety of the bath surface is generally excited at all times. 22…”

“…Durey et al 21 developed a relatively sophisticated method to analyze the drop's trajectory in terms of stable subtrajectories, which allowed the authors to demonstrate that the double quantisation of orbital radius and angular momentum occurs even in the high-memory regime, where the dynamics are dominated by erratic switching between unstable periodic and quasi-periodic sub-trajectories. Kurianski et al 22 revisited this system theoretically with the stroboscopic model of Oza et al 23 and captured a number of periodic, quasiperiodic, and chaotic walker trajectories, including a number that were not reported experimentally. Kurianski et al also demonstrated that a requirement for double quantisation in the simple harmonic potential system is that the memory time (the characteristic decay time of the Faraday waves) exceeds the crossing time (the characteristic time taken for the drop to span its maximum range).…”

“…While walking drops in corrals 6,9,10,13,29 and harmonic potentials [17][18][19][20][21][22] are two of the more robust and rich examples of hydrodynamic quantum analogs, to date these two systems have been treated separately. Thinking of the circular corral walls as an alternative means of inducing a radial potential on a walking droplet allows one to place these systems on equal footing.…”

Ratcheting droplet pairs

“…The stroboscopic model of Oza et al 11 has proven to be sufficient for predicting the stability of a variety of singlewalker states, including the bouncing and walking states, as well as circular orbits in a rotating frame 4,11 and in the presence of a simple harmonic potential. 12,13 However, the stroboscopic model cannot be expected to capture the behavior of ratcheting pairs for several reasons. First, the stroboscopic model does not account for variations in a drop's vertical dynamics and so would be unable to capture the range of bouncing modes, including (1, 1), (2,2), and (2, 1) modes, seen in our experiments.…”