It has been shown earlier that in the linear response regime, dephasing by point scatterers (within the self-consistent Born approximation) can be visualized in terms of point voltage probes attached to each space and energy coordinate (r,E). In this paper we derive a generalized linear response equation starting from the non-equilibrium Green function formalism that can be used to describe any dephasing process in any approximation. The dephasing is characterized by a 'reservoir function' which can be evaluated from the self-energy. The linear response equation can be visualized in terms of voltage probes but with individual probes connected to each pair of spatial coordinates and to each energy (r,r',E). Unlike point scatteren, this generalized 'probe' model allows us to introduce phase relaxation without necessarily introducing momentum relaxation. We obtain explicit expressions for the transmission 'Tij from terminal 'j' to terminal 'i' by eliminating the 'floating probes' inside the device. These expressions for Tij clearly show the role of the exclusion principle in determining the transmission. Proof of reciprocity in multiterminal conductors is provided. We also present a simple illustrative example calculating Tij for a short single-moded electron waveguide with electron-phonon interactions. An important difference between the present formulation and usual linear response theory is that the electrochemical potential difference is treated as the driving force; however, we do not neglect the self-consistent fields that appear in an interacting system when a small bias is applied.
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