  We consider a finite quantum system under slow driving and weakly coupled to
thermal reservoirs at different temperatures. We present a systematic
derivation of the quantum master equation for the density matrix and the
out-of-time-order correlators. We start from the microscopic Hamiltonian and we
formulate the equations ruling the dynamics of these quantities by recourse to
the Schwinger-Keldysh non-equilibrium Green's function formalism, performing a
perturbative expansion in the coupling between the system and the reservoirs.
We focus on the adiabatic dynamics, which corresponds to considering the linear
response in the ratio between the relaxation time due to the system-reservoir
coupling and the time scale associated to the driving. We calculate the
particle and energy fluxes. We illustrate the formalism in the case of a qutrit
coupled to bosonic reservoirs and of a pair of interacting quantum dots
attached to fermionic reservoirs, also discussing the relevance of coherent
effects.
