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arXiv:2408.13761 · Physical review. E

Tunneling in a Lorenz-like model for an active wave-particle entity

Runze Xu, Rahil N. Valani

Published Mar 25, 2025• 1 citation
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Abstract

Active wave-particle entities (WPEs) emerge as self-propelled oil droplets on the free surface of a vibrating oil bath. The particle (droplet) periodically imprints decaying waves on the liquid surface which in turn guide the particle motion, resulting in a two-way coupling between the particle and its self-generated waves. Such WPEs have been shown to exhibit hydrodynamic analogs of various quantum features. In this work, we theoretically and numerically explore a dynamical analog of tunneling by considering a simple setup of a one-dimensional WPE incident on an isolated Gaussian potential barrier. Our idealized model takes the form of a perturbed Lorenz system, which we use to explore the dynamics and statistics of barrier crossing as a function of initial conditions and system parameters. Our work highlights that velocity fluctuations of the WPE at high memories that are rooted in nonequilibrium features of the Lorenz system, such as spiraling motion toward equilibrium points and transient chaos, give rise to (i) sensitivity and unpredictability in barrier crossing, (ii) smooth variations in transmission probability as a function of system parameters, and (iii) wave-like features in the transmitted and reflected probability density profiles.

DOI
10.1103/physreve.111.034218

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Active wave-particle entities (WPEs) emerge as self-propelled oil droplets on the free surface of a vibrating oil bath. The particle (droplet) periodically imprints decaying waves on the liquid surface which in turn guide the particle motion, resulting in a two-way coupling between the particle and its self-generated waves. Such WPEs have been shown to exhibit hydrodynamic analogs of various quantum features. In this work, we theoretically and numerically explore a dynamical analog of tunneling by considering a simple setup of a one-dimensional WPE incident on an isolated Gaussian potential barrier. Our idealized model takes the form of a perturbed Lorenz system, which we use to explore the dynamics and statistics of barrier crossing as a function of initial conditions and system parameters. Our work highlights that velocity fluctuations of the WPE at high memories that are rooted in nonequilibrium features of the Lorenz system, such as spiraling motion toward equilibrium points and transient chaos, give rise to (i) sensitivity and unpredictability in barrier crossing, (ii) smooth variations in transmission probability as a function of system parameters, and (iii) wave-like features in the transmitted and reflected probability density profiles.
A macroscopic, self-propelled wave–particle entity (WPE) that emerges as a walking droplet on the surface of a vibrating liquid bath exhibits several hydrodynamic quantum analogs. We explore the rich dynamical and quantum-like features emerging in a model of an idealized one-dimensional WPE in a double-well potential. The integro-differential equation of motion for the WPE transforms to a Lorenz-like system, which we explore in detail. We observe the analog of quantized eigenstates as discrete limit cycles that arise by varying the width of the double-well potential, and also in the form of multistability with coexisting limit cycles. These states show narrow as well as wide energy level splitting. Tunneling-like behavior is also observed where the WPE erratically transitions between the two wells of the double-well potential. We rationalize this phenomena in terms of crisis-induced intermittency. Further, we discover a fractal structure in the escape time distribution of the particle from a well based on initial conditions, indicating unpredictability of this tunneling-like intermittent behavior at all scales. The chaotic intermittent dynamics lead to wave-like emergent features in the probability distribution of particle’s position that show qualitative similarity with its quantum counterpart. Lastly, rich dynamical features are also observed such as a period doubling route to chaos as well as self-similar periodic islands in the chaotic parameter set.
A macroscopic, self-propelled wave-particle entity (WPE) that emerges as a walking droplet on the surface of a vibrating liquid bath exhibits several hydrodynamic quantum analogs. We explore the rich dynamical and quantum-like features emerging in a model of an idealized one-dimensional wave-particle entity in a double-well potential. The integro-differential equation of motion for the WPE transforms to a Lorenz-like system, which we explore in detail. We observe the analog of quantized eigenstates as discrete limit cycles that arise by varying the width of the double-well potential, and also in the form of multistability with coexisting limit cycles. These states show narrow as well as wide energy level splitting. Tunneling-like behavior is also observed where the WPE erratically transitions between the two wells of the double-well potential. We rationalize this phenomena in terms of crisis-induced intermittency. Further, we discover a fractal structure in the escape time distribution of the particle from a well based on initial conditions, indicating unpredictability of this tunneling-like intermittent behavior at all scales. The chaotic intermittent dynamics lead to wave-like emergent features in the probability distribution of particle's position that show qualitative similarity with its quantum counterpart. Lastly, rich dynamical features are also observed such as a period doubling route to chaos as well as self-similar periodic islands in the chaotic parameter set.
Active particles are nonequilibrium 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.
A droplet bouncing on the surface of a vertically vibrating liquid bath can walk horizontally, guided by the waves it generates on each impact. This results in a self-propelled classical particle-wave entity. By using a one-dimensional theoretical pilot-wave model with a generalized wave form, we investigate the dynamics of this particle-wave entity. We employ different spatial wave forms to understand the role played by both wave oscillations and spatial wave decay in the walking dynamics. We observe steady walking motion as well as unsteady motions such as oscillating walking, self-trapped oscillations, and irregular walking. We explore the dynamical and statistical aspects of irregular walking and show an equivalence between the droplet dynamics and the Lorenz system, as well as making connections with the Langevin equation and deterministic diffusion.
A droplet bouncing on a vibrated bath becomes a "walker" moving at constant velocity on the interface when it couples to the surface wave it generates. Here the motion of a walker is investigated when it collides with barriers of various thicknesses. Surprisingly, it undergoes a form of tunneling: the reflection or transmission of a given incident walker is unpredictable. However, the crossing probability decreases exponentially with increasing barrier width. This shows that this wave-particle association has a nonlocality sufficient to generate a quantumlike tunneling at a macroscopic scale.
2009)] presented experimental results demonstrating the unpredictable tunneling of a classical wave-particle association as may arise when a droplet walking across the surface of a vibrating fluid bath approaches a submerged barrier.We here present a theoretical model that captures the influence of bottom topography on this wave-particle association and so enables us to investigate its interaction with barriers.The coupled wave-droplet dynamics results in unpredictable tunneling events.As reported in the experiments by Eddi et al. and as is the case in quantum tunneling [Gamow, Nature (London) 122, 805 (1928)], the predicted tunneling probability decreases exponentially with increasing barrier width.In the parameter regimes examined, tunneling between two cavities suggests an underlying stationary ergodic process for the droplet's position.