Joule expansion of a pure many-body state
  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. 
S. Camalet