  We calculate the real time non-equilibrium dynamics of quantum spin systems
at finite temperatures. The mathematical framework originates from the
$C^*$-approach to quantum statistical mechanics and is applied to samples
investigated by means of spin-polarized scanning tunneling microscopy. Quantum
fluctuations around thermal equilibrium are analyzed and calculated. The time
averaged expectation values agree with the time averaged experimental data for
magnetization curves. The method is used to investigate the dynamics of a
sample for shorter times than the resolution time of the experimental setup.
Furthermore, predictions for relaxation times of single spins on metallic and
semiconductor surfaces are made. To check the validity of our model we compare
our results with experimental data obtained from Fe adatoms on InSb and Co
adatoms on Pt(111) and find good agreement. Approximated thermalization is
found numerically for the expectation values of the spin operators.
