# Hard disk model

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

where is the intermolecular pair potential between two disks at a distance , and 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[edit]

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 and the hexatic phase at , and a continuous transition between the hexatic and solid phases at ^{[5]}. Note that the maximum possible packing fraction is given by ^{[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: ) ^{[9]}. Also studied via integral equations ^{[10]}.
Experimental results ^{[11]}.

## Equations of state[edit]

*Main article: Equations of state for hard disks*

## Virial coefficients[edit]

*Main article: Hard sphere: virial coefficients*

## See also[edit]

## References[edit]

- ↑ 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) - ↑ 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) - ↑ B. J. Alder and T. E. Wainwright "Phase Transition in Elastic Disks", Physical Review
**127**pp. 359-361 (1962) - ↑ C. H. Mak "Large-scale simulations of the two-dimensional melting of hard disks", Physical Review E
**73**065104(R) (2006) - ↑ E. P. Bernard and W. Krauth "Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition", Physical Review Letters
**107**155704 (2011) - ↑ L. Fejes Tóth "Über einen geometrischen Satz." Mathematische Zeitschrift
**46**pp. 83-85 (1940) - ↑ M. Engel, J. A. Anderson, S. C. Glotzer, M. Tsobe, E. P. Bernard, and W. Krauth "Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods", Physical Review E
**87**042134 (2013) - ↑ Jarosław Piasecki, Piotr Szymczak, and John J. Kozak "Prediction of a structural transition in the hard disk fluid", Journal of Chemical Physics
**133**164507 (2010) - ↑ John J. Kozak, Jack Brzezinski and Stuart A. Rice "A Conjecture Concerning the Symmetries of Planar Nets and the Hard disk Freezing Transition", Journal of Physical Chemistry B
**112**pp. 16059-16069 (2008) - ↑ Luis Mier-y-Terán, Brian Ignacio Machorro-Martínez, Gustavo A. Chapela, and Fernando del Río "Study of the hard-disk system at high densities: the fluid-hexatic phase transition", Journal of Chemical Physics
**148**234502 (2018) - ↑ Alice L. Thorneywork, Joshua L. Abbott, Dirk G. A. L. Aarts, and Roel P. A. Dullens "Two-Dimensional Melting of Colloidal Hard Spheres", Physical Review Letters
**118**158001 (2017)

**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[edit]

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