  The mixing process of discrete-time quantum walks on one-dimensional lattices
is revisited in a setting where the walker is coupled to an environment, and
the time evolution of the walker and the environment is unitary. The mixing
process is found to be incomplete, in the sense that the walker does not
approach the maximally mixed state indefinitely, but the distance to the
maximally mixed state saturates to some finite value depending on the size of
the environment. The quantum speedup of mixing time is investigated numerically
as the size of the environment decreases from infinity to a finite value. The
mixing process in this unitary setting can be explained by interpreting it as
an equilibration process in a closed quantum system, where subsystems can
exhibit equilibration even when the entropy of the total system remains zero.
