Lorenz-like systems emerging from an integro-differential trajectory equation of a one-dimensional wave-particle entity

TL;DR

This paper transforms an integro-differential equation describing a wave-particle entity into Lorenz-like systems, revealing chaotic dynamics and offering new insights into hydrodynamic quantum analogs and physical interpretations of chaos.

Contribution

It introduces a method to derive Lorenz-like systems from wave-particle dynamics, linking hydrodynamic phenomena with chaos theory in a novel way.

Findings

Lorenz-like systems emerge from wave-particle equations for various wave forms

Chaotic dynamics can be modeled by transformed ODE systems

Provides a new physical interpretation of Lorenz-like chaos

Abstract

Vertically vibrating a liquid bath can give rise to a self-propelled wave-particle entity on its free surface. The horizontal walking dynamics of this wave-particle entity can be described adequately by an integro-differential trajectory equation. By transforming this integro-differential equation of motion for a one-dimensional wave-particle entity into a system of ordinary differential equations (ODEs), we show the emergence of Lorenz-like dynamical systems for various spatial wave forms of the entity. Specifically, we present and give examples of Lorenz-like dynamical systems that emerge when the wave form gradient is (i) a solution of a linear homogeneous constant coefficient ODE, (ii) a polynomial and (iii) a periodic function. Understanding the dynamics of the wave-particle entity in terms of Lorenz-like systems may provide to be useful in rationalizing emergent statistical behavior from underlying chaotic dynamics in hydrodynamic quantum analogs of walking droplets. Moreover, the results presented here provide an alternative physical interpretation of various Lorenz-like dynamical systems in terms of the walking dynamics of a wave-particle entity.