Infinite-memory classical wave-particle entities, attractor-driven active particles and the diffusionless Lorenz equations

TL;DR

This paper models a classical wave-particle entity in a high-memory regime, showing that its complex dynamics can be described by diffusionless Lorenz equations, revealing rich chaotic behaviors and connections to hydrodynamic quantum analogs.

Contribution

It introduces a simplified 3D ODE model (DLEs) for infinite-memory wave-particle entities, linking phase-space attractors to their chaotic and statistical behaviors.

Findings

WPE dynamics reduce to diffusionless Lorenz equations in the infinite-memory limit.

The DLE system exhibits diverse periodic and chaotic behaviors.

Phase-space analysis links attractors to WPE motion characteristics.

Abstract

A classical wave-particle entity (WPE) can materialize as a millimeter-sized droplet walking horizontally on the free surface of a vertically vibrating liquid bath. This WPE comprises a particle (droplet) that shapes its environment by locally exciting decaying standing waves, which in turn guides the particle motion. At high amplitude of bath vibrations, the particle-generated waves decay very slowly in time and the particle motion is influenced by the history of waves along its trajectory. In this high-memory regime, WPEs exhibit hydrodynamic quantum analogs where quantum-like statistics arise from underlying chaotic dynamics. Exploration of WPE dynamics in the very high-memory regime requires solving an integro-differential equation of motion. By using an idealized one-dimensional WPE model where the particle generates sinusoidal waves, we show that in the limit of infinite memory, the system dynamics reduce to a $3$D nonlinear system of ordinary differential equations (ODEs) known as the diffusionless Lorenz equations (DLEs). We use our algebraically simple ODE system to explore in detail, theoretically and numerically, the rich set of periodic and chaotic dynamical behaviors exhibited by the WPE in the parameter space. Specifically, we link the geometry and dynamics in phase-space of the DLE system to the dynamical and statistical features of WPE motion, paving a way to understand hydrodynamic quantum analogs using phase-space attractors. Our system also provides an alternate interpretation of an attractor-driven particle, i.e.~an active particle driven by internal state-space variables of the DLE system. Hence, our results might also provide new insights in modeling active particle locomotion.