Hard disk model

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

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

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 . In a recent publication by Mak 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 $\eta =\pi /{\sqrt {12}}\approx 0.906899...$ ) . Similar results have been found using the BBGKY hierarchy and by studying tessellations (the hexatic region: $0.680<\eta <0.729$) .

Equations of state

Main article: Equations of state for hard disks

Virial coefficients

Main article: Hard sphere: virial coefficients

References

Related reading

• Ya G Sinai "Dynamical systems with elastic reflections", Russian Mathematical Surveys 25 pp. 137-189 (1970)
• Katherine J. Strandburg, John A. Zollweg, and G. V. Chester "Bond-angular order in two-dimensional Lennard-Jones and hard-disk systems", Physical Review B 30 pp. 2755 - 2759 (1984)
• Nándor Simányi "Proof of the Boltzmann-Sinai ergodic hypothesis for typical hard disk systems", Inventiones Mathematicae 154 pp. 123-178 (2003)
• Roland Roth, Klaus Mecke, and Martin Oettel "Communication: Fundamental measure theory for hard disks: Fluid and solid", Journal of Chemical Physics 136 081101 (2012)

External links

• Hard disks and spheres computer code on SMAC-wiki.