Hamiltonian of mean force for damped quantum systems

We consider a quantum system linearly coupled to a reservoir of harmonic oscillators. For finite coupling strengths, the stationary distribution of the damped system is not of the Gibbs form, in contrast to standard thermodynamics. With the help of the quantum Hamiltonian of mean force, we quantify this deviation exactly for a harmonic oscillator and provide approximations in the limit of high and low temperatures, and weak and strong couplings. Moreover, in the semiclassical regime, we use the quantum Smoluchowski equation to obtain results valid for any potential. We, finally, give a physical interpretation of the deviation in terms of the initial system-reservoir coupling.