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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 | 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>(\rho \leq 0.890)</math> hexatic <math>(\rho > 0.920)</math> solid. | ||
==Equations of state== | ==Equations of state== | ||
:''Main article: [[Equations of state for hard disks]]'' | :''Main article: [[Equations of state for hard disks]]'' | ||
==Virial coefficients== | ==Virial coefficients== | ||
:''Main article: [[Hard sphere: virial coefficients]]'' | :''Main article: [[Hard sphere: virial coefficients]]'' | ||
==References== | ==References== | ||
<references/> | <references/> | ||
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*[http://dx.doi.org/10.1103/PhysRevB.30.2755 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)] | *[http://dx.doi.org/10.1103/PhysRevB.30.2755 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)] | ||
*[http://dx.doi.org/10.1007/s00222-003-0304-9 Nándor Simányi "Proof of the Boltzmann-Sinai ergodic hypothesis for typical hard disk systems", Inventiones Mathematicae '''154''' pp. 123-178 (2003)] | *[http://dx.doi.org/10.1007/s00222-003-0304-9 Nándor Simányi "Proof of the Boltzmann-Sinai ergodic hypothesis for typical hard disk systems", Inventiones Mathematicae '''154''' pp. 123-178 (2003)] | ||
==External links== | ==External links== | ||
*[http://www.smac.lps.ens.fr/index.php/Programs_Chapter_2:_Hard_disks_and_spheres Hard disks and spheres] computer code on SMAC-wiki. | *[http://www.smac.lps.ens.fr/index.php/Programs_Chapter_2:_Hard_disks_and_spheres Hard disks and spheres] computer code on SMAC-wiki. | ||
[[Category: Models]] | [[Category: Models]] |