We reinterpret the microcanonical conditions in the quantum domain as constraints for the interaction of the “gas subsystem” under consideration and its environment (“container”). The time average of a purity measure is found to equal the average over the respective path in Hilbert space. We then show that for typical (degenerate or nondegenerate) thermodynamical systems almost all states within the allowed region of Hilbert space have a local von Neumann entropy $S$ close to the maximum and a purity $P$ close to its minimum, respectively. Typically, thermodynamical systems should obey the second law.

• Received 14 August 2000

DOI:https://doi.org/10.1103/PhysRevLett.86.1927