  We study the influence of quenched disorder on quantum phase transitions in
systems with over-damped dynamics. For Ising order parameter symmetry disorder
destroys the sharp phase transition by rounding because a static order
parameter can develop on rare spatial regions. This leads to an exponential
dependence of the order parameter on the coupling constant. At finite
temperatures the static order on the rare regions is destroyed. This restores
the phase transition and leads to a double-exponential relation between
critical temperature and coupling strength. We discuss the behavior based on
Lifshitz-tail arguments and illustrate the results by simulations of a model
system.
