Quantum Transport Equations in Nanoelectronics

[asheg]

Hi Folks,

In the quantum transport, specially in calculating nano transistors' characteristics, we use a set of equations to find electron's density and specteral function. I want to know where they came from. The equations are :

[G]: Green's function matrix
[A(E)]: Spectral function
f1: fermi function in source
f2: fermi function in drain
H: Hamiltonian

[G] = [EI - H - \Sigma_{1} - \Sigma_{2} ]^{-1}

Equ1: [A(E)] = i [G - G\dagger]

Equ2: [G^{n}(E)] = [G \Gamma_{1} G \dagger] f1 + [G \Gamma_{2} G\dagger] f2

Could anyone say how I can prove these equations from Green's function definition?

Thanks