  In this work a two-particle irreducible (2PI) closed-time-path (CTP)
effective action is used to describe the nonequilibrium dynamics of a Bose
Einstein condensate (BEC) selectively loaded into every third site of a
one-dimensional optical lattice. The motivation of this work is the recent
experimental realization of this system at National Institute of Standards and
Technology (NIST) where the placement of atoms in an optical lattice is
controlled by using an intermediate superlattice. Under the 2PI CTP scheme with
this initial configuration, three different approximations are considered: a)
the Hartree-Fock-Bogoliubov (HFB) approximation, b) the next-to-leading order
1/$\mathcal{N}$ expansion of the 2PI effective action up to second order in the
interaction strength and c) a second order perturbative expansion in the
interaction strength. We present detailed comparisons between these
approximations and determine their range of validity by contrasting them with
the exact many body solution for a moderate number of atoms and wells. As a
general feature we observe that because the second order 2PI approximations
include multi-particle scattering in a systematic way, they are able to capture
damping effects exhibited in the exact solution that a mean field collisionless
approach fails to produce. While the second order approximations show a clear
improvement over the HFB approximation our numerical result shows that they do
not work so well at late times, when interaction effects are significant.
