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We treat the problem of the approach to thermal equilibrium by only resortingto quantum dynamics of an isolated macroscopic system. Inspired by the twoimportant works in 2009 and in 1929, we have noted that a condition we call"thermodynamic normality" for a macroscopic observable guarantees the approachto equilibrium (in the sense that a measurement of the observable at time $t$almost certainly yields a result close to the corresponding microcanonicalaverage for a sufficiently long and typical $t$). A crucial point is that wemake no assumptions on the initial state of the system, except that its energyis distributed close to a certain macroscopic value. We also present three(rather artificial) models in which the thermodynamic normality can beestablished, thus providing concrete examples in which the approach toequilibrium is rigorously justified. Note that this kind of results which holdfor ANY initial state are never possible in classical systems. We are thusdealing with a mechanism which is peculiar to quantum systems. The present noteis written in a self-contained (and hopefully readable) manner. It onlyrequires basic knowledge in quantum physics and equilibrium statisticalmechanics.

The approach to thermal equilibrium and "thermodynamic normality" --- An  observation based on the works by Goldstein, Lebowitz, Mastrodonato, Tumulka,  and Zanghi in 2009, and by von Neumann in 1929