A pair of oscillators interacting with a common heat bath

TL;DR

This paper investigates the interactions of two oscillators coupled to a common heat bath, exploring how their separation affects their dynamics and fluctuation forces using a generalized Lamb model.

Contribution

It extends the Lamb model to analyze two oscillators interacting via a shared heat bath, addressing the effects of their separation on their behavior.

Findings

Derived coupled Langevin equations for the oscillators

Obtained limits for large and small separations

Calculated fluctuation forces at different separations

Abstract

Here the problem considered is that of a pair of oscillators coupled to a common heat bath. Many, if not most, discussions of a single operator coupled to a bath have used the independent oscillator model of the bath. However, that model has no notion of separation, so the question of phenomena when the oscillators are near one another compared with when they are widely separated cannot be addressed. Here the Lamb model of an oscillator attached to a stretched string is generalized to illustrate some of these questions. The coupled Langevin equations for a pair of oscillators attached to the string at different points are derived and their limits for large and small separations obtained. Finally, as an illustration of a different phenomenon, the fluctuation force between a pair of masses attached to the string is calculated, with closed form expressions for the force at small and large separations.