How do we use non-equilibrium Green's functions (NEGF) or the Keldysh formalism in the theory of quantum transport? Please take a simple example like the Hopping model with a non-equilibrium perturbation that couples two lattice sites for example, with increased hopping strength between those two particular lattice sites and explain in that context. Show with this simple example how we use the NEGF or the Keldysh formalism.
I would like to point out that Lecture 1 in the lecture entitled "Field Theoretical Methods for Non-equilibrium Transport Phenomena" in the following link: http://tfp1.physik.uni-freiburg.de/eu_www/Miraflores/ contains exactly the solution to the simple example (and more!) that you want to know.