Effects of Weak Dissipation on the Long-Time Behaviour of the Quantized Standard Map

The dynamics generated by the quantized standard map with weak dissipation is studied numerically on long time-scales. It is found that even extremely weak damping is capable of disrupting the localization in the action variable which suppresses chaotic motion in the conservative quantized standard map, and of restoring diffusion of action. The results are interpreted on the basis of the analogy, originally developed for the conservative map, to Anderson localization in disordered solids.