  We investigate the asymptotic state of time-periodic quantum systems with
regular and chaotic Floquet states weakly coupled to a heat bath. The
asymptotic occupation probabilities of these two types of states follow
fundamentally different distributions. Among regular states the probability
decreases from the state in the center of a regular island to the outermost
state by orders of magnitude, while chaotic states have almost equal
probabilities. We derive an analytical expression for the occupations of
regular states of kicked systems, which depends on the winding numbers of the
regular tori and the parameters temperature and driving frequency. For a
constant winding number within a regular island it simplifies to Boltzmann-like
weights $\exp(-\betaeff \Ereg_m)$, similar to time-independent systems. For
this we introduce the regular energies $\Ereg_m$ of the quantizing tori and an
effective winding-number-dependent temperature $1/\betaeff$, different from the
actual bath temperature. Furthermore, the occupations of other typical Floquet
states in a mixed phase space are studied, i.e. regular states on nonlinear
resonances, beach states, and hierarchical states, giving rise to distinct
features in the occupation distribution. Avoided crossings involving a regular
state lead to drastic consequences for the entire set of occupations. We
introduce a simplified rate model whose analytical solutions describe the
occupations quite accurately.
