  We develop a theory for the nonequilibrium coherent transport through a
mesoscopic region, based on the nonequilibrium Green function technique. The
theory requires the weak coupling between the central mesoscopic region and the
multiple electrodes connected to it, but allows arbitrary hopping and
interaction in the central region. An equation determining the nonequilibrium
distribution in the central interacting region is derived and plays an
important role in the theory. The theory is applied to two special cases for
demonstrations, revealing the novel effects associated with the combination of
phase coherence, Coulomb interaction, and nonequilibrium distribution.
