Distribution of local entropy in the Hilbert space of bi-partite quantum systems: origin of Jaynes' principle

For a closed bi-partite quantum system partitioned into system proper and environment we interpret the microcanonical and the canonical condition and the constraints for the interaction between those two subsystem. In both cases the possible pure-state trajectories are confined to certain regions in Hilbert space. We show that in a properly defined thermodynamical limit almost all stated within those accessible regions represent states of some maximum local entropy. For the microcanonical condition this dominant state still depends on the initial state: for the canonical condition it coincides with that defined by Jaynes' principle. It is these states which thermodynamical systems should generically evolve into.