  We study the Bose and Fermi Hubbard model in the (formal) limit of large
coordination numbers $Z\gg1$. Via an expansion into powers of $1/Z$, we
establish a hierarchy of correlations which facilitates an approximate
analytical derivation of the time-evolution of the reduced density matrices for
one and two sites etc. With this method, we study the quantum dynamics
(starting in the ground state) after a quantum quench, i.e., after suddenly
switching the tunneling rate $J$ from zero to a finite value, which is still in
the Mott regime. We find that the reduced density matrices approach a (quasi)
equilibrium state after some time. For one lattice site, this state can be
described by a thermal state (within the accuracy of our approximation).
However, the (quasi) equilibrium state of the reduced density matrices for two
sites including the correlations cannot be described by a thermal state. Thus,
real thermalization (if it occurs) should take much longer time. This behavior
has already been observed in other scenarios and is sometimes called
``pre-thermalization.'' Finally, we compare our results to numerical
simulations for finite lattices in one and two dimensions and find qualitative
agreement.
