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 feel at very close distances.

Formal definition

Hard spheres of diameter $\backslash sigma$ are particles with the following pairwise interaction potential:

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

Virial coefficients

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

{| $\backslash frac\{B\_2\}\{v\_0\}$=$4\{\backslash frac\{\}\{\}\}$ $\backslash frac\{B\_3\}$ $\backslash frac\{B\_4\}\{35\; \backslash pi\}+\backslash frac\{4131\}\{35\; \backslash pi\}\; \backslash arccos\{\backslash frac\{1\}\{\backslash sqrt\{3\}\}\}\backslash approx\; 18.365$

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

{| $\backslash frac\{B\_5\}\{\{v\_0\}^4\}$= $28.24\; \backslash pm\; 0.08$ $\backslash frac\{B\_6\}\{\{v\_0\}^5\}$= $39.5\; \backslash pm\; 0.4$ $\backslash frac\{B\_7\}\{\{v\_0\}^6\}$= $56.5\; \backslash pm\; 1.6$

Literature

• J. P. Hansen and I. R. McDonald Theory of Simple Liquids Academic Press, London (1986)

Statistical mechanicsSpecific models