  We investigate implications of decoherence for quantum systems which are
classically chaotic. We show that, in open systems, the rate of von Neumann
entropy production quickly reaches an asymptotic value which is: (i)
independent of the system-environment coupling, (ii) dictated by the dynamics
of the system, and (iii) dominated by the largest Lyapunov exponent. These
results shed a new light on the correspondence between quantum and classical
dynamics as well as on the origins of the ``arrow of time.''
