  We obtain exact analytic expressions for a class of functions expressed as
integrals over the Haar measure of the unitary group in d dimensions. Based on
these general mathematical results, we investigate generic dynamical properties
of complex open quantum systems, employing arguments from ensemble theory. We
further generalize these results to arbitrary eigenvalue distributions,
allowing a detailed comparison of typical regular and chaotic systems with the
help of concepts from random matrix theory. To illustrate the physical
relevance and the general applicability of our results we present a series of
examples related to the fields of open quantum systems and nonequilibrium
quantum thermodynamics. These include the effect of initial correlations, the
average quantum dynamical maps, the generic dynamics of system-environment pure
state entanglement and, finally, the equilibration of generic open and closed
quantum systems.
