Non-Hamiltonian features of a classical pilot-wave dynamics

TL;DR

This paper models the non-Hamiltonian dynamics of a bouncing droplet in a harmonic potential as a Rayleigh oscillator, providing theoretical insights that match experimental observations and enabling future wave-particle interaction studies.

Contribution

It introduces a simple Rayleigh-type friction model for a walker in a harmonic potential, capturing its non-Hamiltonian behavior and stability properties.

Findings

The model accurately predicts the walker’s dynamics.

Attractors and stability are characterized analytically.

Results agree well with experimental data.

Abstract

A bouncing droplet on a vibrated bath can couple to the waves it generates, so that it becomes a propagative walker. Its propulsion at constant velocity means that a balance exists between the permanent input of energy provided by the vibration and the dissipation. Here we seek a simple theoretical description of the resulting non-Hamiltonian dynamics with a walker immersed in a harmonic potential well. We demonstrate that the interaction with the recently emitted waves can be modeled by a Rayleigh-type friction. The Rayleigh oscillator has well defined attractors. The convergence toward them and their stability is investigated through an energetic approach and a linear stability analysis. These theoretical results provide a description of the dynamics in excellent agreement with the experimental data. It is thus a basic framework for further investigations of wave-particle interactions when memory effects are included.