  Homogenization protocols model the quantum mechanical evolution of a system
to a fixed state independently from its initial configuration by repeatedly
coupling it with a collection of identical ancillas. Here we analyze these
protocols within the formalism of "relaxing" channels providing an easy to
check sufficient condition for homogenization. In this context we describe
mediated homogenization schemes where a network of connected qudits relaxes to
a fixed state by only partially interacting with a bath. We also study
configurations which allow us to introduce entanglement among the elements of
the network. Finally we analyze the effect of having competitive configurations
with two different baths and we prove the convergence to dynamical equilibrium
for Heisenberg chains.
