Quantum-mechanical extension of the Lebowitz-Penrose theorem on the Van Der Waals theory

Recently Lebowitz and Penrose gave a rigorous derivation of the van der Waals-Maxwell theory of the liquid-vapor transition, and showed how the Maxwell equal area-rule could be obtained from a proper statistical mechanical calculation. Their results are quite general - being valid in any number of dimensions and for a broad class of pair potentials - but they were proved only for classical mechanics. In the present work we extend the proof to quantum systems with any statistics - Boltzmann, Bose, or Fermi. One corollary of this extended theorem is a model of a Bose gas with a first-order phase transition.