Hard disk model

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

${\mathrm{\Phi }}_{12}(r)=\{\begin{array}{ccc}\mathrm{\infty }& ;& r<\sigma \\ 0& ;& r\ge \sigma \end{array}$

where ${\mathrm{\Phi }}_{12}(r)$ 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: $0.680<\eta <0.729$) . Also studied via integral equations . Experimental results .

Equations of state

Main article: Equations of state for hard disks

Virial coefficients

Main article: Hard sphere: virial coefficients

See also

• Binary hard-disk mixtures

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.