  In this paper, we look at four generalizations of the one dimensional
Aubry-Andre-Harper (AAH) model which possess mobility edges. We map out a phase
diagram in terms of population imbalance, and look at the system size
dependence of the steady state imbalance. We find non-monotonic behaviour of
imbalance with system parameters, which contradicts the idea that the
relaxation of an initial imbalance is fixed only by the ratio of number of
extended states to number of localized states. We propose that there exists
dimensionless parameters, which depend on the fraction of single particle
localized states, single particle extended states and the mean participation
ratio of these states. These ingredients fully control the imbalance in the
long time limit and we present numerical evidence of this claim. Among the four
models considered, three of them have interesting duality relations and their
location of mobility edges are known. One of the models (next nearest neighbour
coupling) has no known duality but mobility edge exists and the model has been
experimentally realized. Our findings are an important step forward to
understanding non-equilibrium phenomena in a family of interesting models with
incommensurate potentials.
