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 . Recent works show a phase diagram containing an isotropic, a hexatic, and a solid phase . 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 $\eta =0.700$ and the hexatic phase at $\eta =0.716$, and a continuous transition between the hexatic and solid phases at $\eta =0.720$ . Note that the maximum possible packing fraction is given by $\eta =\pi /{\sqrt {12}}\approx 0.906899...$ . This scenario has since been confirmed using a variety of simulation methods .

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.