  We consider the generic model of a finite-size quantum electron system
connected to two (temperature and particle) reservoirs. The quantum open system
is driven out of equilibrium by the presence of both a temperature and a
chemical potential differences between the two reservoirs. The nonequilibrium
(NE) thermodynamical properties of such a quantum open system are studied for
the steady state regime. In such a regime, the corresponding NE density matrix
is built on the so-called generalised Gibbs ensembles. From different
expressions of the NE density matrix, we can identify the terms related to the
entropy production in the system. We show, for a simple model, that the entropy
production rate is always a positive quantity. Alternative expressions for the
entropy production are also obtained from the Gibbs-von Neumann conventional
formula and discussed in detail. Our results corroborate and expand earlier
works found in the literature.
