# Stepwise relaxation and stochastic precession in degenerate oscillators dispersively coupled to particles

## TL;DR

This paper investigates the relaxation dynamics of degenerate oscillators coupled to particles, revealing a stepwise energy dissipation process influenced by particle trapping and thermal noise, with implications for understanding dispersive interactions.

## Contribution

It introduces a novel analysis of stepwise relaxation in dispersively coupled oscillators, highlighting the role of particle trapping and stochastic precession in energy dissipation.

## Key findings

- Stepwise relaxation characterized by alternating fast and slow dissipation phases.
- Thermal noise induces stochastic precession of mode mixing angles.
- Stepwise relaxation observed only with thermal noise in the membrane model.

## Abstract

By numerical integration, we study the relaxation dynamics of degenerate harmonic oscillator modes dispersively coupled to particle positions. Depending on whether the effective inertial potential induced by the oscillators keep the particles confined, or if the particle trajectories traverse the system, the local oscillator energy dissipation rate changes drastically. The inertial trapping, release and retrapping of particles results in a characteristic step-wise relaxation process, with alternating regions of fast and slow dissipation. To demonstrate this phenomenon we consider first a one-dimensional minimal prototype model which displays these characteristics. We then treat the effect of dispersive interaction in a model corresponding to an adsorbate diffusing on a circular membrane interacting with its three lowest vibrational modes. In the latter model, stepwise relaxation appears only in the presence of thermal noise, which also causes a slow-in-time stochastic precession of the mixing angle between the degenerate eigenmodes.