Deviation from equilibrium in macroscopic quantum systems

The irreversible relaxation to equilibrium is explained for macroscopic quantum systems with an emphasis on the behavior of expectation values and extremely high dimensionality of the Hilbert space. We consider a large but isolated system that is initially out of equilibrium and eventually relaxes to equilibrium. The relaxation is described by the deviation of the expectation value of the quantity of interest from the long-time average. After relaxation, the amount of deviation from equilibrium is discussed based on probabilistic arguments, which are available for nonintegrable systems. We also evaluate how long the system stays near equilibrium.