Hard disk model: Difference between revisions
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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. 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>(\ | Despite the apparent simplicity of this model/system, the phase behaviour and the nature of the phase transitions remains an area of active study. 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>. | ||
==Equations of state== | ==Equations of state== | ||
Revision as of 14:50, 19 October 2011
Hard disks are hard spheres in two dimensions. The hard disk intermolecular pair potential is given by[1] [2]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}\left( r \right) = \left\{ \begin{array}{lll} \infty & ; & r < \sigma \\ 0 & ; & r \ge \sigma \end{array} \right. }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}\left(r \right) } is the intermolecular pair potential between two disks at a distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r := |\mathbf{r}_1 - \mathbf{r}_2|} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma } 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
Despite the apparent simplicity of this model/system, the phase behaviour and the nature of the phase transitions remains an area of active study. In a recent publication by Mak [3] using over 4 million particles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2048^2)} one appears to have the phase diagram isotropic Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\eta < 0.699)} , a hexatic phase, and a solid phase Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\eta > 0.723)} (the maximum possible packing fraction is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta = \pi / \sqrt{12} \approx 0.906899...} [4]) . Similar results have been found using the BBGKY hierarchy [5] and by studying tessellations (the hexatic region: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.680 < \eta < 0.729} ) [6].
Equations of state
- Main article: Equations of state for hard disks
Virial coefficients
- Main article: Hard sphere: virial coefficients
References
- ↑ 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)
- ↑ 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)
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)
External links
- Hard disks and spheres computer code on SMAC-wiki.