Entropy production in quantium brownian motion

I investigate how to coherently define entropy production for a process of transient relaxation in the Quantum Brownian Motion model for the harmonic potential. I compare a form, called ``Poised" (P), which after non-Markovian transients corresponds to a definition of heat as the change in the system Hamiltonian of mean force, with a recent proposal by Esposito et al (ELB) based on a definition of heat as the energy change in the bath. Both expressions yield a positive-defined entropy production and coincide for vanishing coupling strength, but their difference is proved to be always positive (after non-Markovian transients disappear) and to grow as the coupling strength increases. In the classical over-damped limit the ``Poised" entropy production converges to the entropy production used in stochastic thermodynamics. We also investigate the effects of the system size, and of the ensuing Poincarà© recurrences, and how the classical limit is approached. I close by discussing the strong-coupling limit, in which the ideal canonical equilibrium of the bath is violated.