  We derive the Joule expansion of an isolated perfect gas from the principles
of quantum mechanics. Contrary to most studies of irreversible processes which
consider composite systems, the gas many-body Hilbert space cannot be
factorised into Hilbert spaces corresponding to interesting and ignored degrees
of freedom. Moreover, the expansion of the gas into the entire accessible
volume is obtained for pure states. Still, the number particle density is
characterised by a chemical potential and a temperature. We discuss the special
case of a boson gas below the Bose condensation temperature.
