Nonequilibrium density matrix in quantum open systems: Generalization for simultaneous heat and charge steady-state transport

We suggest a generalization of the expression of the nonequilibrium (NE) density matrix obtained by Hershfield's method for the cases where both heat and charge steady-state currents are present in a quantum open system. The finite-size quantum system, connected to two temperature and particle reservoirs, is driven out of equilibrium by the presence of both a temperature gradient and a chemical potential gradient between the two reservoirs. We show that the NE density matrix is given by a generalized Gibbs-like ensemble and is in full agreement with the general results of the McLennan-Zubarev nonequilibrium ensembles. The extra nonequilibrium terms are related to the entropy production in the system and characterize the fluxes of heat and particle. An explicit example, for the lowest-order expansion, is provide for a model system of noninteracting fermions.