  A system composed of identical spins and described by a quantum mechanical
pure state is analyzed within the statistical framework presented in Part I of
this work. We explicitly derive the typical values of the entropy, of the
energy, and of the equilibrium reduced density matrix of a subsystem for the
two different statistics introduced in Part I. In order to analyze their
consistency with thermodynamics, these quantities of interest are evaluated in
the limit of large number of components of the isolated system. The main
results can be summarized as follows: typical values of the entropy and of the
equilibrium reduced density matrix as functions of the internal energy in the
fixed expectation energy ensemble do not satisfy the requirement of
thermodynamics. On the contrary, the thermodynamical description is recovered
from the random pure state ensemble (RPSE), provided that one considers systems
large enough. The thermodynamic limit of the considered properties for the spin
system reveals a number of important features. First canonical statistics (and
thus, canonical typicality as long as the fluctuations around the average value
are small) emerges without the need of assuming the microcanonical space for
the global pure state. Moreover, we rigorously prove (i) the equivalence of the
"global temperature," derived from the entropy equation of state, with the
"local temperature" determining the canonical state of the subsystems; and (ii)
the equivalence between the RPSE typical entropy and the canonical entropy for
the overall system.
