Hard spheres

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Hard spheres are widely used as model particles in the statistical mechanical theory of fluids and solids. They are defined simply as impenetrable spheres that cannot overlap in space. They mimic the extremely strong repulsion that atoms and spherical molecules experience at very close distances.

Formal definition

Hard spheres of diameter $σ$ are particles with the following pairwise interaction potential:

$V(\mathbf{r}_1,\mathbf{r}_2)=\left\{ \begin{matrix}0 & \mbox{if}\quad |\mathbf{r}_1-\mathbf{r}_2| \geq \sigma \\ \infty & \mbox{if}\quad|\mathbf{r}_1-\mathbf{r}_2| < \sigma \end{matrix} \right.$

where $\mathbf{r}_1$ and $\mathbf{r}_2$ are the positions of the two particles.

Virial coefficients

The first three virial coefficients for hard spheres can be determined analytically

$\frac{B_2}{v_0}$	=	$4{\frac{}{}}$
$\frac{B_3}{{v_0}^2}$	=	$10{\frac{}{}}$
$\frac{B_4}{{v_0}^3}$	=	$-\frac{712}{35}+\frac{219 \sqrt{2}}{35 \pi}+\frac{4131}{35 \pi} \arccos{\frac{1}{\sqrt{3}}}\approx 18.365$

Higher-order ones can be determined numerically using Monte Carlo integration. We list

$\frac{B_5}{{v_0}^4}$	=	$28.24 \pm 0.08$
$\frac{B_6}{{v_0}^5}$	=	$39.5 \pm 0.4$
$\frac{B_7}{{v_0}^6}$	=	$56.5 \pm 1.6$

A table of virial coefficients for up to eight dimensions can be found on the page Hard sphere: virial coefficients.

Literature

J. P. Hansen and I. R. McDonald Theory of Simple Liquids Academic Press, London (1986)
Hard sphere model page on SklogWiki.

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