  We consider small subsystems of large, closed quantum systems that evolve
according to the von Neumann equation. Without approximations and without
making any special assumptions on the form of the interaction we prove that,
for almost all initial states and almost all times, the off-diagonal elements
of the density matrix of the subsystem in the eigenbasis of its local
Hamiltonian must be small, whenever the energy difference of the corresponding
eigenstates is larger than the interaction energy. This proves that decoherence
with respect to the local energy eigenbasis is a natural property of weakly
interacting quantum systems.
