  Many theoretical and experimental results show that even isolated quantum
systems evolving unitarily may equilibrate, since the evolution of some
observables may be around an equilibrium value with negligible fluctuations
most of the time. There are rigorous theorems giving the conditions for such
equilibration to happen. In particular, initial states prepared with a lack of
resolution in the energy will equilibrate. We investigate how equilibration may
be affected by a lack of resolution, or coarse-graining, in the magnetization
of the initial state. In particular, for a chaotic spin chain and using exact
diagonalization, we show that the level of equilibration of an initial state
with a coarse, not well-defined magnetization is different from the level of an
initial state with well-defined magnetization. This difference will depend on
the degree of coarse-graining and the direction of magnetization. We also
analyze the time for the system to reach equilibrium, showing good agreement
with theoretical estimates and with some evidence that less resolution leads to
faster equilibration. Our study highlights the crucial role of initial state
preparation in the equilibration dynamics of quantum systems and provides new
insights into the fundamental nature of equilibration in closed systems.
