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

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... g 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.