  Evidently, some relaxation dynamics, e.g. exponential decays, are much more
common in nature than others. Recently there have been attempts to explain this
observation on the basis of ``typicality of perturbations'' with respect to
their impact on expectation value dynamics. These theories suggest that a
majority of the very numerous, possible Hamiltonian perturbations entail more
or less the same type of alteration of the decay dynamics. Thus, in this paper,
we study how the approach towards equilibrium in closed quantum systems is
altered due to weak perturbations. To this end, we perform numerical
experiments on a particular, exemplary spin system. We compare our numerical
data to predictions from three particular theories. We find satisfying
agreement in the weak perturbation regime for one of these approaches.
