# Difference between revisions of "Hard disk model"

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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>. 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>. | + | 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>. Also studied via [[integral equations]] <ref>[https://doi.org/10.1063/1.5026496 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)]</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>. | 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>. | ||

## Latest revision as of 11:45, 25 June 2018

**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.

## Contents

## 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]}. In a recent publication by Mak ^{[4]} 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 ^{[5]}) . Similar results have been found using the BBGKY hierarchy ^{[6]} and by studying tessellations (the hexatic region: ) ^{[7]}. Also studied via integral equations ^{[8]}.
Experimental results ^{[9]}.

## 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) - ↑ 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)

**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.