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In this work a two-particle irreducible (2PI) closed-time-path (CTP)effective action is used to describe the nonequilibrium dynamics of a BoseEinstein condensate (BEC) selectively loaded into every third site of aone-dimensional optical lattice. The motivation of this work is the recentexperimental realization of this system at National Institute of Standards andTechnology (NIST) where the placement of atoms in an optical lattice iscontrolled by using an intermediate superlattice. Under the 2PI CTP scheme withthis initial configuration, three different approximations are considered: a)the Hartree-Fock-Bogoliubov (HFB) approximation, b) the next-to-leading order1/$\mathcal{N}$ expansion of the 2PI effective action up to second order in theinteraction strength and c) a second order perturbative expansion in theinteraction strength. We present detailed comparisons between theseapproximations and determine their range of validity by contrasting them withthe exact many body solution for a moderate number of atoms and wells. As ageneral feature we observe that because the second order 2PI approximationsinclude multi-particle scattering in a systematic way, they are able to capturedamping effects exhibited in the exact solution that a mean field collisionlessapproach fails to produce. While the second order approximations show a clearimprovement over the HFB approximation our numerical result shows that they donot work so well at late times, when interaction effects are significant.Comment: 34 pages, 7 figure

Nonequilibrium dynamics of optical-lattice-loaded Bose-Einstein-condensate atoms: Beyond the Hartree-Fock-Bogoliubov approximation