  In this paper, we consider the Hamiltonian evolution of N weakly interacting
Bosons. Assuming triple collisions, its mean field approximation is given by a
quintic Hartree equation. We construct a second order correction to the mean
field approximation using a kernel k(t,x,y) and derive an evolution equation
for k. We show the global existence for the resulting evolution equation for
the correction and establish an apriori estimate comparing the approximation to
the exact Hamiltonian evolution. Our error estimate is global and uniform in
time. Comparing with the work in [22,12,13] where the error estimate grows in
time, our approximation tracks the exact dynamics for all time with an error of
the order O(1/$\sqrt{N}$).
