A Liouville equation for systems which exchange particles with   reservoirs: transport through a nano-device
  A Redfield-like Liouville equation for an open system that couples to one or more leads and exchanges particles with them is derived. The equation is presented for a general case. A case study of time-dependent transport through a single quantum level for varying electrostatic and chemical potentials in the leads is presented. For the case of varying electrostatic potentials the proposed equation yields, for the model study, the results of an exact solution. 
Igor V. Ovchinnikov and Daniel Neuhauser