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.

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. In a recent publication by Mak (Ref. 5) using over 4 million particles $(2048^{2})$ one appears to have the phase diagram isotropic $(\rho \leq 0.890)$ hexatic $(\rho >0.920)$ solid.

Equations of state

Main article: Equations of state for hard disks

Virial coefficients

Main article: Hard sphere: virial coefficients

References

1. Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller and Edward Teller, "Equation of State Calculations by Fast Computing Machines", Journal of Chemical Physics 21 pp.1087-1092 (1953)
2. W. W. Wood "Monte Carlo calculations of the equation of state of systems of 12 and 48 hard circles", Los Alamos Scientific Laboratory Report LA-2827 (1963)
3. Ya G Sinai "Dynamical systems with elastic reflections", Russian Mathematical Surveys 25 pp. 137-189 (1970)
4. 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)
5. Nándor Simányi "Proof of the Boltzmann-Sinai ergodic hypothesis for typical hard disk systems", Inventiones Mathematicae 154 pp. 123-178 (2003)
6. C. H. Mak "Large-scale simulations of the two-dimensional melting of hard disks", Physical Review E 73 065104(R) (2006)

External links

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