We use the scattering matrix approach to derive generalized Bardeen-like formulae for the conductances between the contacts of a phase-coherent multiprobe conductor and a tunneling tip which probes its surface. These conductances are proportional to local partial densities of states, called injectivities and emissivities. The current and the current fluctuations measured at the tip are related to an effective local non-equilibrium distribution function. This distribution function contains the quantum- mechanical phase-coherence of the charge carriers in the conductor and is given as products of injectivities and the Fermi distribution functions in the electron reservoirs. The results are illustrated for measurements on ballistic conductors with barriers and for diffusive conductors.
Gramespacher, T., & Büttiker, M. (1999). Local non-equilibrium distribution of charge carriers in a phase- coherent conductor. Comptes Rendus de l’Academie de Sciences - Serie IIb: Mecanique, Physique, Chimie, Astronomie, 327(9), 877–884. https://doi.org/10.1016/S1287-4620(99)80150-1