  We investigate Bose-Hubbard chains in a central depleted well configuration,
with dephasing in the middle well. We look at equilibration of populations,
pseudo-entropy, and entanglement measures. Using stochastic integration in the
truncated Wigner representation, we find that the initial quantum states of the
occupied wells has an influence on the subsequent dynamics, and that with more
than three wells, the chains do not reach a full equilibrium, with edge effects
becoming important, and the time to reach the steady state becoming longer. The
evolutions with and without phase diffusion are qualitatively different. We
find no convincing evidence of entanglement in the final states with phase
diffusion. Although at least one accepted measure indicates the presence of
mode entanglement, we are easily able to show that it can give ambiguous
predictions.
