Active wave-particle clusters

TL;DR

The paper investigates how memory-coupled active wave-particle droplets form bound clusters that exhibit a nucleus-like confinement and a rich spectrum of collective modes. Using a stroboscopic integrodifferential framework with a damped, oscillatory wave field $W(|\mathbf{x}|)=\cos(|\mathbf{x}|)\exp[-(|\mathbf{x}|/L)^2]$, the authors map the parameter space of decay length $L$ and memory $\tau$ to identify stationary states, regular excitations (breathing, quadrupole, surface, compression, rotating), chaotic dynamics, and decay processes. They uncover that the time-averaged wave field induces a common mean-field-like potential across modes, drawing qualitative parallels with nuclear shell and bag models, and observe exponential-like cluster decay reminiscent of radioactive processes. These results provide a classical hydrodynamic platform to explore quantum-like phenomena, confinement, and stochastic decay in active matter, with implications for designing and interpreting experiments on interacting walking droplets. Overall, the study reveals how memory-mediated interactions drive self-organization into nucleus-like clusters that support discrete excitations and memory-driven decay, offering insights into hydrodynamic analogs of nuclear physics.

Abstract

Active particles are non-equilibrium entities that uptake energy and convert it into self-propulsion. A dynamically rich class of inertial active particles having features of wave-particle coupling and wave memory are walking/superwalking droplets. Such classical, active wave-particle entities (WPEs) have previously been shown to exhibit hydrodynamic analogs of many single-particle quantum systems. Inspired by the rich dynamics of strongly interacting superwalking droplets in experiments, we numerically investigate the dynamics of WPE clusters using a stroboscopic model. We find that several interacting WPEs self-organize into a stable bound cluster, reminiscent of an atomic nucleus. This active cluster exhibits a rich spectrum of collective excitations, including shape oscillations and chiral rotating modes, akin to vibrational and rotational modes of nuclear excitations, as the spatial extent of the waves and their temporal decay rate (memory) are varied. Dynamically distinct excitation modes create a common time-averaged collective wave field potential, bearing qualitative similarities with the nuclear shell model and the bag model of hadrons. For high memory and rapid spatial decay of waves, the active cluster becomes unstable and disintegrates; however, within a narrow regime of the parameter space, the cluster ejects single particles whose decay statistics follow exponential laws, reminiscent of radioactive nuclear decay. Our study uncovers a rich spectrum of dynamical behaviors in clusters of active particles, opening new avenues for exploring hydrodynamic quantum analogs in active matter systems.

Figures (12)

• Figure 1: Active wave-particle cluster. A finite-size particle (gray) generates (a) an axisymmetric wave of form $W(|\mathbf{x}|)=\cos(|\mathbf{x}|)\exp[{-(|\mathbf{x}|/L)^2}]$ (peaks in red and troughs in blue) at each instant of time. (b) The superposition of all the individual waves generated by the particle results in an overall wave field that propels an isolated particle with constant velocity (black arrow), making it active. (c) Simulated snapshot in time of a collection of such active WPEs that self-organize into a "nucleus" structure, where the particles stay bound by their self-generated collective wave field comprising a potential well walled by a wave barrier. (d) Experimental image of a self-organized cluster of superwalking droplets superwalker.
• Figure 2: Dynamical regimes for active wave-particle clusters having $N=10$ particles. Different dynamical behaviors for the active wave-particle cluster in the parameter space formed by the dimensionless spatial decay length scaled with the Faraday wavelength, $L/2\pi$, and the dimensionless memory parameter, $\tau$, for small particles of dimensionless diameter $a=1.01$ with $R=0.77$ (upper part of the figure) and larger particles of $a=1.63$ with $R=1.72$ (lower part of the figure). The white dashed curve in both parts denotes the critical memory curve $\tau_c=1/\sqrt{{R\left(1+2L^{-2}\right)}}$, above which an isolated WPE transitions from a stationary state to a steady walking state. Different dynamical regimes are color coded as follows: unbound clusters (yellow), stationary clusters (red), chaotic excitations (purple), particle ejection (pink), breathing mode (teal), quadrupole mode (maroon), surface oscillations (blue), compression mode (cyan), intermittent rotating mode (bright green) and rotating mode (dark green).
• Figure 3: Stationary clusters of active WPEs at low memory. A cluster of $N=10$ WPEs self-organize into various different static configurations based on the size of each particle $a$ and the length scale of spatial decay $L$ at low values of the memory parameter $\tau$. Panels (a)-(e) and (f)-(j) show configurations for two different particle sizes $a=1.01$ and $a=1.63$, respectively, for spatial decay (respective values from left to right for both top and bottom panels) $L/2\pi=0.20, 0.49, 0.78, 1.07, 1.50$. Other parameters are $\tau=0.17$ and $R=0.77$ for panels (a)-(e), and $\tau=0.44$ and $R=1.72$ for (f)-(j).
• Figure 4: Collective excitation modes of active WPE clusters. A cluster of $N=10$ WPEs each with size (a)-(b) $a=1.01$, $R=0.77$ and (c)-(e) $a=1.63$, $R=1.72$, display several collective excitation modes that include: (a) breathing mode (teal, $\tau=1.13$, $L/2\pi=1.5$) where the cluster stays circular (white dashed curve) and the cluster radius $\langle R_c \rangle$ varies periodically, (b) quadrupole mode (maroon, $\tau=5.91$, $L/2\pi=1.5$) where the cluster oscillates between two orthogonally oriented ellipses (black solid and white dashed ovals with maroon line indicating inclination of the ellipse), (c) surface oscillations (blue, $\tau=1.03$, $L/2\pi=0.78$) where the cluster stays circular (black solid oval) and the particles on the surface oscillate azimuthally, (d) compression mode (cyan, $\tau=1.03$, $L/2\pi=1.07$) where the cluster oscillates between a circular and an elliptical shape (black solid and white dashed ovals) and the particles on the surface undergo radial oscillations, and (e) rotating mode (dark green, $\tau=3.37$, $L/2\pi=1.07$) where the ellipse maintains a constant eccentricity (black solid oval) and its inclination (green line) rotates at a constant rate. See Supplemental Videos S1-S5 supplementary_m for videos of modes in (a)-(e).
• Figure 5: Statistics of collective excitation modes for a cluster of $N=10$ WPEs. (a)-(e) Probability distribution showing (bottom) time-averaged particle density and (top) the time-averaged collective wave field $\bar{h}(x,y)$, corresponding to the excitation modes in Fig. \ref{['Fig: Cluster excitations']}. Panel (f) shows the mean cluster kinetic energy $\bar{E}$ (solid horizontal lines) within their time-averaged collective potential $\bar{h}(x,0)$ along a horizontal centerline for different modes for small particle size in (a)-(b), whereas (g) shows the same for large particle size excitation modes in (c)-(e). Shaded regions indicate the uncertainty in the $\bar{E}$ calculated as the standard deviation. Note, also, that the color scheme for all lines in panels (f) and (g) correspond to the excitation-mode labels in panels (a)-(e).