  We derive a general expression for the electron nonequilibrium (NE)
distribution function in the context of steady state quantum transport through
a two-terminal nanodevice with interaction. The central idea for the use of NE
distributions for open quantum systems is that both the NE and many-body (MB)
effects are taken into account in the statistics of the finite size system
connected to reservoirs. We develop an alternative scheme to calculate the NE
steady state properties of such systems. The method, using NE distribution and
spectral functions, presents several advantages, and is equivalent to
conventional steady-state NE Green's functions (NEGF) calculations when the
same level of approximation for the MB interaction is used. The advantages of
our method resides in the fact that the NE distribution and spectral functions
have better analytic behaviour for numerical calculations. Furthermore our
approach offer the possibility of introducing further approximations, not only
at the level of the MB interaction as in NEGF, but also at the level of the
functional form used for the NE distributions. For the single level model with
electron-phonon coupling we have considered, such approximations provide a good
representation of the exact results, for either the NE distributions themselves
or the transport properties. We also derive the formal extensions of our method
for systems consisting of several electronic levels and several vibration
modes.
