in the non-equilibrium

$$G_C^<(t) = i \langle c_i^+ c_j (t) \rangle$$

One must calculate

$$\rho_{ij} = \langle c_i^+ c_j(0) \rangle$$ (90)

$$= \frac{1}{i} G_C^{<}(0)$$ (91)

$$= \frac{1}{2\pi i} \int_{-\infty}^{\infty} d\omega G_{C}^{<} (\omega )$$ (92)

This can be evaluated only approximately.

Because ,

$$2\pi A = i (G^R-G^A) = G^R\Gamma G^A = G^R (\Gamma_R+\Gamma_L) G^A$$ (93)

$A$ is the spectrum function.

At region C,

$$2\pi A = G^R\Gamma G^A = G^R (\Gamma_R+\Gamma_L) G^A$$ (94)

where $\Gamma=\Gamma_R+\Gamma_L$ because $\Sigma=\Sigma_R +\Sigma_L$.

Assume that the region of C near the region L and R is close to the equilibrium. If not, the self-energy defined in the region C can not be evaluated.

$$G_{C}^{<} = G_C^R \Sigma^< G_C^A$$ (95)

$$\simeq G_C^R i (\Gamma_L f(\omega -\mu_L) + \Gamma_R f(\omega -\mu_R) ) G_C^A$$ (96)

$$= - G_C^R ( (\Sigma_L^R-\Sigma_L^A) f(\omega -\mu_L) + (\Sigma_R^R - \Sigma_R^A) f(\omega -\mu_R) ) G_C^A$$ (97)

Note that the left side of eq.(98) is not analytical.

If $\omega < \min(\mu_L,\mu_R)-10T$, $f(\omega -\mu)=1$. then

$$G^< = 2 \pi i A$$ (98)

then eq.(93) is

$$\rho_{ij} = \int_{-\infty}^{E_1} d\omega A_{ij}(w) + \int_{E_1}^{\infty} \frac{d\omega }{2\pi i} G_{ij}^<(\omega )$$ (99)

The first term can be calculated in the same manner as the case of the equilibrium condition. For $\omega \ll\mu$, the electronic structure of the system will be close to the equilibrium. $E_1$ ( $< \min(\mu_L,\mu_R)-10T$) is the energy below which the electronic structure is close enough to the equilibrium state. Note $G^<(\omega )$ goes 0 for $\omega >\max(\mu_L,\mu_R)+10T$. the upper limit of the second term can be not an infinite.