Vacuum and thermal energies for two oscillators interacting through a field

TL;DR

This paper models the Casimir-Polder interaction between two oscillators coupled to a scalar field, incorporating dissipation via heat baths, and derives thermodynamic properties including free energy and Matsubara representation.

Contribution

It introduces a first principles approach to include dissipation in a simple oscillator-field system and derives new expressions for free energy and thermodynamic consistency.

Findings

Derived an expression for free energy in terms of real frequencies.

Obtained the Matsubara representation with dissipation.

Showed the zeroth Matsubara frequency contribution is modified without violating thermodynamics.

Abstract

We consider a simple (1+1)-dimensional model for the Casimir-Polder interaction consisting of two oscillators coupled to a scalar field. We include dissipation in a first principles approach by allowing the oscillators to interact with heat baths. For this system, we derive an expression for the free energy in terms of real frequencies. From this representation, we derive the Matsubara representation for the case with dissipation. Further we consider the case of vanishing intrinsic frequencies of the oscillators. We show that in this case the contribution from the zeroth Matsubara frequency gets modified and no problems with the laws of thermodynamics appear.