# Hard disk model

Hard disks are hard spheres in two dimensions. The hard disk intermolecular pair potential is given by[1] [2]

$\Phi_{12}\left( r \right) = \left\{ \begin{array}{lll} \infty & ; & r < \sigma \\ 0 & ; & r \ge \sigma \end{array} \right.$

where $\Phi_{12}\left(r \right)$ is the intermolecular pair potential between two disks at a distance $r := |\mathbf{r}_1 - \mathbf{r}_2|$, and $\sigma$ is the diameter of the disk. This page treats hard disks in a two-dimensional space, for three dimensions see the page hard disks in a three dimensional space.

## Phase transitions

Despite the apparent simplicity of this model/system, the phase behaviour and the nature of the phase transitions remains an area of active study ever since the early work of Alder and Wainwright [3]. In a recent publication by Mak [4] using over 4 million particles $(2048^2)$ one appears to have the phase diagram isotropic $(\eta < 0.699)$, a hexatic phase, and a solid phase $(\eta > 0.723)$ (the maximum possible packing fraction is given by $\eta = \pi / \sqrt{12} \approx 0.906899...$ [5]) . Similar results have been found using the BBGKY hierarchy [6] and by studying tessellations (the hexatic region: $0.680 < \eta < 0.729$) [7]. Also studied via integral equations [8]. Experimental results [9].

## Equations of state

Main article: Equations of state for hard disks

## Virial coefficients

Main article: Hard sphere: virial coefficients