  A recently proposed Markov approach provides Lindblad-type scattering
superoperators, which ensure the physical (positive-definite) character of the
many-body density matrix. We apply the mean-field approximation to such
many-body equation, in the presence of one- and two-body scattering mechanisms,
and we derive a closed equation of motion for the electronic single-particle
density matrix, which turns out to be non-linear as well as non-Lindblad. We
prove that, in spite of its nonlinear and non-Lindblad structure, the
mean-field approximation does preserve the positive-definite character of the
single-particle density matrix, an essential prerequisite of any reliable
kinetic treatment of semiconductor quantum devices. This result is in striking
contrast with conventional (non-Lindblad) Markov approaches, where the
single-particle mean-field equations can lead to positivity violations and to
unphysical results. Furthermore, the proposed single-particle formulation is
extended to the case of quantum systems with spatial open boundaries, providing
a formal derivation of a recently proposed density-matrix treatment based on a
Lindblad-like system-reservoir scattering superoperator.
