Dissipative dynamics of a Harmonic Oscillator : A non-perturbative approach

TL;DR

This paper develops a non-perturbative microscopic approach to derive a master equation for a harmonic oscillator interacting with a bath, analyzing its dissipative dynamics and thermodynamic properties.

Contribution

It introduces a non-perturbative derivation of the master equation for a harmonic oscillator, extending previous methods used for free Brownian particles.

Findings

Analytical expressions for diffusion constants.

Positivity of the master equation above a critical temperature.

Recovery of free Brownian particle results in the zero-frequency limit.

Abstract

Starting from a microscopic theory, we derive a master equation for a harmonic oscillator coupled to a bath of non-interacting oscillators. We follow a non-perturbative approach, proposed earlier by us for the free Brownian particle. The diffusion constants are calculated analytically and the positivity of the Master Equation is shown to hold above a critical temperature. We compare the long time behaviour of the average kinetic and potential energies with known thermodynamic results. In the limit of vainishing oscillator frequency of the system, we recover the results of the free Brownian particle.