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Quantum master equations are common tools to describe the dynamics of many-body systems open to an environment. Due to the interaction with the latter, even for the case of noninteracting electrons, the computational cost to solve these equations increases exponentially with the partical number. We propose a simple scheme, which allows to study the dynamics of N noninteracting electrons taking into account both dissipation effects and Fermi statistics, with a computational cost that scales linearly with N. Our method is based on a mapping of the many-body system to a specific set of effective single-particle systems. We provide detailed numerical results showing excellent agreement between the effective single-particle scheme and the exact many-body one, as obtained from studying the dynamics of two different systems. In the first, we study optically-induced currents in quantum rings at zero temperature, and in the second we study a linear chain coupled at its ends to two thermal baths with different (finite) temperatures. In addition, we give an analytical justification for our method, based on an exacting averaging over the many-body states of the original master equations.

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