Vacuum and Thermal Energies for Two Oscillators Interacting Through A Field

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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.