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

Researchers have long sought classical systems that mimic quantum behaviors. While quantum phenomena often rely on probabilistic wave functions, classical systems can replicate similar statistical outcomes through deterministic chaos. In this context, wave-particle entities (WPEs) have emerged as promising candidates. These systems involve droplets that bounce on vibrating fluid surfaces, generating waves that guide their motion. Prior studies have shown that such droplets can exhibit behaviors analogous to quantum particles, such as tunneling and interference. However, the mechanisms behind these analogs remain unclear. Specifically, the role of memory in wave decay and how it influences droplet dynamics has not been fully explored. This gap motivated the current work, which focuses on high-memory regimes where wave decay is minimal. In such conditions, droplet motion becomes history-dependent, leading to complex dynamical patterns. No prior work had resolved how these patterns relate to attractor-based dynamics in phase space. This paper addresses that uncertainty by examining the transition to infinite memory in WPE systems.

The study aims to explore the behavior of wave-particle entities (WPEs) in the infinite-memory regime, where wave decay is negligible. The researchers sought to understand how the droplet's motion is influenced by its own wave history and how this leads to quantum-like statistical patterns. The motivation stems from the need to connect classical chaotic dynamics with observed quantum analogs in WPE systems. By modeling the droplet as a particle that generates and interacts with waves, the authors aimed to derive a simplified mathematical framework that captures the system's dynamics. They focused on the transition from finite to infinite memory, which occurs when wave decay approaches zero. The study also aimed to investigate how the resulting equations relate to known nonlinear systems, such as the diffusionless Lorenz equations (DLEs). The researchers hypothesized that the DLEs could describe the droplet's motion in this regime. This approach allows for a deeper understanding of the underlying phase-space attractors and their role in shaping statistical behaviors.

The researchers developed a one-dimensional model of a wave-particle entity (WPE) where the droplet generates sinusoidal waves. They derived an integrodifferential equation of motion that accounts for the particle's interaction with its own wave field. By taking the limit of infinite memory, where wave decay is negligible, the equation simplifies to a system of three ordinary differential equations (ODEs). These ODEs correspond to the diffusionless Lorenz equations (DLEs), a well-known nonlinear system in dynamical systems theory. The model assumes that the droplet's motion is influenced by the cumulative effect of past wave interactions. The authors used numerical simulations to explore the behavior of the DLE system across different parameter values. They analyzed the phase-space geometry of the system to identify attractors and their influence on droplet dynamics. The study also compared the statistical properties of the droplet's motion with those observed in high-memory WPE experiments. The researchers focused on periodic and chaotic behaviors, linking them to the structure of the DLE phase space. This approach allowed them to examine how attractor geometry influences the droplet's trajectory and statistical outcomes.

The study found that in the infinite-memory regime, the droplet's motion is governed by the diffusionless Lorenz equations (DLEs). These equations describe a system with three variables and exhibit both periodic and chaotic dynamics depending on parameter values. Numerical simulations revealed that the DLE system supports a wide range of behaviors, including limit cycles and strange attractors. The researchers observed that the droplet's trajectory in phase space corresponds to these attractors, which determine the statistical properties of its motion. The system's behavior closely matches experimental observations of high-memory WPEs, where droplets exhibit quantum-like statistics. The study also identified specific parameter ranges where the droplet's motion transitions from periodic to chaotic. These transitions were linked to bifurcations in the DLE system, such as period-doubling and crisis events. The researchers found that the droplet's motion becomes increasingly sensitive to initial conditions as memory increases. This sensitivity leads to complex statistical patterns that resemble those seen in quantum systems. The study further showed that the droplet's motion is not random but is instead shaped by the underlying attractor structure. These findings suggest that the DLE system provides a useful framework for understanding the dynamics of high-memory WPEs.

The authors conclude that the diffusionless Lorenz equations (DLEs) accurately describe the motion of wave-particle entities (WPEs) in the infinite-memory regime. They propose that the droplet's trajectory is determined by the attractors of the DLE system, which shape its statistical behavior. The study suggests that the DLE system captures the essential features of high-memory WPE dynamics, including both periodic and chaotic motion. The researchers emphasize that the droplet's motion is not random but is instead governed by the structure of the phase space. They also note that the DLE system provides a bridge between classical chaotic dynamics and observed quantum analogs in WPE systems. The study further implies that attractor-driven motion offers a new perspective on active particle locomotion. The authors suggest that the DLE system could be used to model other active particles that are influenced by internal state variables. The findings support the idea that complex statistical patterns in WPE systems arise from deterministic attractor dynamics. The study does not claim that the DLE system is the only model for WPE motion but proposes that it is a useful and accurate framework for understanding high-memory regimes.