We consider externally driven many-body systems with complex spectra o
f intrinsic states. The effects of the coupling to the external world
are analyzed by assuming time-dependent random matrix properties for t
he intrinsic system. We derive and solve evolution equations for intri
nsic state population probabilities, average excitation energy and its
fluctuations. The diffusive process is likely to be dominated by memo
ry effects and an unexpected type of fluctuations.