  We analyze closed one-dimensional chains of weakly coupled many level
systems, by means of the so-called Hilbert space average method (HAM). Subject
to some concrete conditions on the Hamiltonian of the system, our theory
predicts energy diffusion with respect to a coarse-grained description for
almost all initial states. Close to the respective equilibrium we investigate
this behavior in terms of heat transport and derive the heat conduction
coefficient. Thus, we are able to show that both heat (energy) diffusive
behavior as well as Fourier's law follows from and is compatible with a
reversible Schroedinger dynamics on the complete level of description.
