STICKY SPHERES IN QUANTUM MECHANICS
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).