  We present the Born-Markov approximated Redfield quantum master equation
(RQME) description for an open system of non-interacting particles (bosons or
fermions) on an arbitrary lattice of $N$ sites in any dimension and weakly
connected to multiple reservoirs at different temperatures and chemical
potentials. The RQME can be reduced to the Lindblad equation, of various forms,
by making further approximations. By studying the $N=2$ case, we show that RQME
gives results which agree with exact analytical results for steady state
properties and with exact numerics for time-dependent properties, over a wide
range of parameters. In comparison, the Lindblad equations have a limited
domain of validity in non-equilibrium. We conclude that it is indeed justified
to use microscopically derived full RQME to go beyond the limitations of
Lindblad equations in out-of-equilibrium systems. We also derive closed form
analytical results for out-of-equilibrium time dynamics of two-point
correlation functions. These results explicitly show the approach to steady
state and thermalization. These results are experimentally relevant for cold
atoms, cavity QED and far-from-equilibrium quantum dot experiments.
