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where <math> \Phi_{12}\left(r \right) </math> is the [[intermolecular pair potential]] between two disks at a distance <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, and <math> \sigma </math> 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]]. | where <math> \Phi_{12}\left(r \right) </math> is the [[intermolecular pair potential]] between two disks at a distance <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, and <math> \sigma </math> 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== | ==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 <ref>[http://dx.doi.org/10.1103/PhysRev.127.359 B. J. Alder and T. E. Wainwright "Phase Transition in Elastic Disks", Physical Review '''127''' pp. 359-361 (1962)]</ref>. | 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 <ref>[http://dx.doi.org/10.1103/PhysRev.127.359 B. J. Alder and T. E. Wainwright "Phase Transition in Elastic Disks", Physical Review '''127''' pp. 359-361 (1962)]</ref>. In a recent publication by Mak <ref>[http://dx.doi.org/10.1103/PhysRevE.73.065104 C. H. Mak "Large-scale simulations of the two-dimensional melting of hard disks", Physical Review E '''73''' 065104(R) (2006)]</ref> using over 4 million particles <math>(2048^2)</math> one appears to have the phase diagram isotropic <math>(\eta < 0.699)</math>, a hexatic phase, and a solid phase <math>(\eta > 0.723)</math> (the maximum possible packing fraction is given by <math>\eta = \pi / \sqrt{12} \approx 0.906899...</math> <ref>[http://dx.doi.org/10.1007/BF01181430 L. Fejes Tóth "Über einen geometrischen Satz." Mathematische Zeitschrift '''46''' pp. 83-85 (1940)]</ref>) . Similar results have been found using the [[BBGKY hierarchy]] <ref>[http://dx.doi.org/10.1063/1.3491039 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)]</ref> and by studying tessellations (the hexatic region: <math>0.680 < \eta < 0.729</math>) <ref>[http://dx.doi.org/10.1021/jp806287e 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)]</ref>. | ||
Similar results have been found using the [[BBGKY hierarchy]] <ref>[http://dx.doi.org/10.1063/1.3491039 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)]</ref> and by studying tessellations (the hexatic region: <math>0.680 < \eta < 0.729</math>) <ref>[http://dx.doi.org/10.1021/jp806287e 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 | |||
Experimental results <ref>[http://dx.doi.org/10.1103/PhysRevLett.118.158001 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)]</ref>. | Experimental results <ref>[http://dx.doi.org/10.1103/PhysRevLett.118.158001 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)]</ref>. | ||