Hard disk model
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.
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 one appears to have the phase diagram isotropic , a hexatic phase, and a solid phase (the maximum possible packing fraction is given by ) . Similar results have been found using the BBGKY hierarchy  and by studying tessellations (the hexatic region: ) . Also studied via integral equations . Experimental results .
Equations of state
- Main article: Equations of state for hard disks
- Main article: Hard sphere: virial coefficients
- 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)
- L. Fejes Tóth "Über einen geometrischen Satz." Mathematische Zeitschrift 46 pp. 83-85 (1940)
- 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)
- 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)
- Hard disks and spheres computer code on SMAC-wiki.