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Many theoretical and experimental results show that even isolated quantumsystems evolving unitarily may equilibrate, since the evolution of someobservables may be around an equilibrium value with negligible fluctuationsmost of the time. There are rigorous theorems giving the conditions for suchequilibration to happen. In particular, initial states prepared with a lack ofresolution in the energy will equilibrate. We investigate how equilibration maybe affected by a lack of resolution, or coarse-graining, in the magnetizationof the initial state. In particular, for a chaotic spin chain and using exactdiagonalization, we show that the level of equilibration of an initial statewith a coarse, not well-defined magnetization is different from the level of aninitial state with well-defined magnetization. This difference will depend onthe degree of coarse-graining and the direction of magnetization. We alsoanalyze the time for the system to reach equilibrium, showing good agreementwith theoretical estimates and with some evidence that less resolution leads tofaster equilibration. Our study highlights the crucial role of initial statepreparation in the equilibration dynamics of quantum systems and provides newinsights into the fundamental nature of equilibration in closed systems.

Equilibration of Isolated Systems: investigating the role of coarse-graining on the initial state magnetization