  Exactly solvable models provide an opportunity to study different aspects of
reduced quantum dynamics in detail. We consider the reduced dynamics of a
single spin in finite XX and XY spin 1/2 chains. First we introduce a general
expression describing the evolution of the reduced density matrix. This
expression proves to be tractable when the combined closed system (i.e. open
system plus environment) is integrable. Then we focus on comparing decoherence
and thermalization timescales in the XX chain. We find that for a single spin
these timescales are comparable, in contrast to what should be expected for a
macroscopic body. This indicates that the process of quantum relaxation of a
system with few accessible states can not be separated in two distinct stages -
decoherence and thermalization. Finally, we turn to finite-size effects in the
time evolution of a single spin in the XY chain. We observe three consecutive
stages of the evolution: regular evolution, partial revivals, irregular
(apparently chaotic) evolution. The duration of the regular stage is
proportional to the number of spins in the chain. We observe a "quiet and cold
period" in the end of the regular stage, which breaks up abruptly at some
threshold time.
