M.D. Penrose, O. Penrose, and G. Stell

For a $3$-dimensional system of hard spheres of diameter $D$ and mass $m$ with an added attractive square-well two-body interaction of width $a$ and depth $\eps$, let $B_{D,a}$ denote the quantum second virial coefficient. Let $B_D$ denote the quantum second virial coefficient for hard spheres of diameter $D$ without the added attractive interaction. We show that in the limit $a \to 0$ at constant $\alpha := \eps m a^2/(2 \hbar ^2)$ with $\alpha < \pi^2/8$,

$$ B_{D,a} = B_{D} - a \left(\frac{\tan \surd (2 \alpha) }{\surd (2 \alpha) }-1 \right) \frac{d}{dD}B_{D} + o(a). $$

The result is true equally for Boltzmann, Bose and Fermi statistics. The method of proof uses the mathematics of Brownian motion. For $\alpha > \pi^2 /8$, we argue that the gaseous phase disappears in the limit $a \to 0$, so that the second virial coefficient becomes irrelevant.

Reviews in Mathematical Physics 6, 947-975 (1994).