  Dynamical universality is the observation that the dynamical properties of
different systems might exhibit universal behavior that are independent of the
system details. In this paper, we study the long-time dynamics of an
one-dimensional noisy quantum magnetic model, and find that even though the
system are inevitably driven to an infinite temperature state, the relaxation
dynamics towards such featureless state can be highly nontrivial and universal.
The effect of various mode-coupling mechanisms (external potential, disorder,
interaction, and the interplay between them) as well as the conservation law on
the long-time dynamics of the systems have been studied, and their relevance
with current ultracold atomic experiments have been discussed.
