Emergence of Friedel-like oscillations from Lorenz dynamics in walking droplets

Abstract

Friedel oscillations are spatially decaying density modulations near localized defects and are a hallmark of quantum systems. Walking droplets provide a macroscopic platform for hydrodynamic quantum analogs, and Friedel-like oscillations were recently observed in droplet-defect scattering through wave-mediated speed modulation [P.~J.~Sáenz \textit{et al.}, \textit{Sci.\ Adv.} \textbf{6}, eay9234 (2020)]. Here we show that Friedel-like oscillatory statistics can also arise from a purely local dynamical mechanism, revealed using a minimal Lorenz model description of a walking droplet viewed as an active particle with internal degrees of freedom. A localized defect directly perturbs the particle's internal dynamical state, generating underdamped velocity oscillations that give rise to oscillatory ensemble position statistics. This work opens new avenues for hydrodynamic quantum analogs by revealing how quantum-like statistics can emerge from local internal-state dynamics of active particles.

Figures (4)

• Figure 1: (a) Lorenz-system model of a walking droplet. A particle at position $x_d$ (blue circle) moves in one dimension with velocity $\dot{x}_d=X$. Its motion is driven by a wave-memory force $Y$ arising from the superposition of exponentially decaying cosine waves generated along its past trajectory (blue field; darker curves denote more recent individual waves), opposed by an effective drag $-X$, and perturbed by an external localized Gaussian barrier (red). The wave height at the particle location is $Z$. (b) Equivalent interpretation as an inertial active particle with position $x_d$ and velocity $X$ interacting with the localized barrier. The particle carries internal ("hidden") variables $Y$ and $Z$, whose dynamics are governed by the Lorenz equations.
• Figure 2: Single-trajectory dynamics in the Lorenz-like walking-droplet model. (a) Space--time trajectory of a particle incident on a localized Gaussian barrier (dashed line at $x_b$, dotted lines at $x_b\pm W$), showing velocity oscillations after interaction; color indicates instantaneous speed $|X|$. (b) Corresponding projection onto the $(X,Y)$ Lorenz phase plane, showing perturbation away from and spiral relaxation toward the steady-walking equilibrium points (black dots). The unstable saddle point at the origin is marked by a red cross. See Supplemental Movie S1 at supplementary_m. Parameters: $R=1$, $\tau=2.5$, $W=1$, $H=5$, $x_b=0$.
• Figure 3: Emergence of Friedel-like ensemble statistics from Lorenz dynamics. Left panels show the evolving position probability density $\Pr(x)$ constructed from an ensemble of $1000$ trajectories, while right panels show the corresponding evolution in the Lorenz phase space projected onto the $(X,Y)$ plane. The plotted distributions $\Pr(x)$ are cumulative probability densities constructed from the ensemble history as time evolves. (a) At $t=0$, the ensemble is localized far from the barrier and clustered (red points) near the steady-walking equilibrium point. (b) At $t=8.2$, interaction with the localized barrier (dashed line at $x_b$, dotted lines at $x_b\pm W$) displaces trajectories away from the equilibrium point, producing transient phase-space excursions and an asymmetric buildup of probability near the barrier. (c) At $t=200$, underdamped spiral relaxation in phase space generates velocity oscillations that translate into spatially oscillatory, Friedel-like modulations of $\Pr(x)$. Black dots denote stable equilibrium points and the red cross denotes the saddle at the origin. See Supplemental Movie S2 at supplementary_m. Parameters: $R=1$, $\tau=2.5$, $W=1$, $H=5$, $x_b=0$.
• Figure 4: Robustness of Friedel-like oscillations to the system dimensionality, wave-field structure and barrier geometry. (a) Emergence of Friedel-like oscillations in a two-dimensional stroboscopic walker model with spatially decaying wave field, where particles initialized far away from a localized bump at the origin are directed towards the center of the bump. The color map shows the long-time probability density $\Pr(x,y)$ of $10000$ outgoing trajectories, revealing concentric oscillatory modulations around the defect. Parameters are $R=1$, $\tau=3$, $W=1$, $H=5$ and $L=2\pi$. (b)--(d) One-dimensional results obtained from the minimal Lorenz-like model for different bump geometries: (b) a steep barrier with $H=10$, (c) a shallow transmissive barrier with $H=1$, and (d) a shallow well with $H=-1$. In all one-dimensional cases, particles are initialized on either side of the defect and directed toward it. In these one-dimensional cases, the plotted distributions $\Pr(x)$ are cumulative probability densities at $t=200$ constructed from the ensemble history as time evolves. Parameters are $R=1$, $\tau=2.5$, $W=1$, $x_b=0$.