  We present a new formulation of the correlated electron-ion dynamics (CEID)
by using equations of motion for nonequilibrium Green's functions, which
generalizes CEID to a general nonequilibrium statistical ensemble that allows
for a variable total number of electrons. We make a rigorous connection between
CEID and diagrammatic perturbation theory, which furthermore allows the key
approximations in CEID to be quantified in diagrammatic terms, and, in
principle, improved. We compare analytically the limiting behavior of CEID and
the self-consistent Born approximation (SCBA) for a general dynamical
nonequilibrium state. This comparison shows that CEID and SCBA coincide in the
weak electron-phonon coupling limit, while they differ in the large ionic mass
limit where we can readily quantify their difference. In particular, we
illustrate the relation between CEID and SCBA by perturbation theory at the
fourth-order in the coupling strength.
