# Hard disk model

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

${\displaystyle \Phi _{12}\left(r\right)=\left\{{\begin{array}{lll}\infty &;&r<\sigma \\0&;&r\geq \sigma \end{array}}\right.}$

where ${\displaystyle \Phi _{12}\left(r\right)}$ is the intermolecular pair potential between two disks at a distance ${\displaystyle r:=|\mathbf {r} _{1}-\mathbf {r} _{2}|}$, and ${\displaystyle \sigma }$ 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]. Recent works show a phase diagram containing an isotropic, a hexatic, and a solid phase [4]. Highly efficient event-chain Monte Carlo simulations of over 1 million hard disks by Bernard and Krauth have solidified this picture, with a first-order phase transition between the fluid at packing fraction ${\displaystyle \eta =0.700}$ and the hexatic phase at ${\displaystyle \eta =0.716}$, and a continuous transition between the hexatic and solid phases at ${\displaystyle \eta =0.720}$ [5]. Note that the maximum possible packing fraction is given by ${\displaystyle \eta =\pi /{\sqrt {12}}\approx 0.906899...}$ [6]. This scenario has since been confirmed using a variety of simulation methods [7].

Similar results have been found using the BBGKY hierarchy [8] and by studying tessellations (the hexatic region: ${\displaystyle 0.680<\eta <0.729}$) [9]. Also studied via integral equations [10]. Experimental results [11].

## Equations of state

Main article: Equations of state for hard disks

## Virial coefficients

Main article: Hard sphere: virial coefficients