  A case study of the non-equilibrium dynamics of open quantum systems in the
markovian approximation is presented for two dynamical models based on a single
harmonic oscillator in an external field. Specified through distinct forms of
ohmic damping, their quantum Langevin equations are derived from an identical
set of physical criteria, namely the canonical commutator between position and
momentum, the Kubo formula, the virial theorem and the quantum equilibrium
variance. The associated Lindblad equations are derived but only one of them is
completely positive. Transforming those into Fokker-Planck equations for the
Wigner functions, both models are shown to evolve towards the same Gibbs state,
for a vanishing external field. The phenomenological differences between the
models are illustrated through their quantum relaxations and through the phase
diagrammes derived from their re-interpretation as mean-field approximations of
an interacting many-body system.
