  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.
