Quantum-Mechanical Extension of the Lebowitz-Penrose Theorem on the Van Der Waals Theory

Abstract

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.