  The Nakajima-Zwanzig(N-Z) equation is a powerful tool to analyze the open
quantum system and non-Markovian behavior. In this paper, we modify the N-Z
equation with assignment maps. This approach extends the application to those
situations in which the bipartite state of the system and environment contains
non-negligible discord initial and/or during the dynamics. We applied the
equation in quantum lattice model and bound the influence of inhomogeneous
terms, which figure out how the dimension of the environment impact the
non-Markovian behavior.
