Non Equilibrium Green's Functions for Dummies: Introduction to the One   Particle NEGF equations
  Non equilibrium Green's function methods are regularly used to calculate current and charge densities in nanoscale (both molecular and semiconductor) conductors under bias. This method is mainly used for ballistic conduction but may be extended to include inelastic scattering. In this tutorial paper the NEGF equations for the current and charge density matrix are derived and explained in a hopefully clear way. 
M. Paulsson