Hard disk model

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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]


  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. B. J. Alder and T. E. Wainwright "Phase Transition in Elastic Disks", Physical Review 127 pp. 359-361 (1962)
  4. C. H. Mak "Large-scale simulations of the two-dimensional melting of hard disks", Physical Review E 73 065104(R) (2006)
  5. E. P. Bernard and W. Krauth "Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition", Physical Review Letters 107 155704 (2011)
  6. L. Fejes Tóth "Über einen geometrischen Satz." Mathematische Zeitschrift 46 pp. 83-85 (1940)
  7. 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)
  8. 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)
  9. 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)
  10. 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)
  11. 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

External links[edit]