Equation of State of a Bose-Einstein System of Particles with Attractive Interactions

Abstract

The equation of state in the grand canonical ensemble is calculated for a system of Bose-Einstein particles with hard-sphere repulsive interactions and weak long-range attractions. The energy levels used in this calculation are modified forms of those derived in an earlier paper. The calculation is carried out in the limit of no interactions, and attention is focused on the thermodynamic phases of the system. It is shown that the gross features of the equation of state of ${\mathrm{He}}^4$ are reproduced. There are the phases: gas, liquid I, and liquid II. The phase transition between gas and the two liquids are first order transitions. The transition terminates in a critical point. The transition between liquid I and liquid II is the Bose-Einstein condensation. Liquid II has a negative coefficient of thermal expansion. Across the transition between liquids I and II the specific heat is discontinuous in value. In the limit of no interactions, the critical point recedes towards zero temperature, zero pressure, and infinite volume.