Hard disk model: Difference between revisions
Jump to navigation
Jump to search
m (Better defined r) |
Carl McBride (talk | contribs) m (Added a reference and a section on the phase transitions.) |
||
Line 9: | Line 9: | ||
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. | 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. | ||
==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. 5) 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]]'' | ||
Line 17: | Line 19: | ||
==References== | ==References== | ||
#[http://dx.doi.org/10.1063/1.1699114 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)] | #[http://dx.doi.org/10.1063/1.1699114 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)] | ||
#[http://dx.doi.org/10.1070/RM1970v025n02ABEH003794 Ya G Sinai "Dynamical systems with elastic reflections", Russian Mathematical Surveys '''25''' pp. 137-189 (1970)] | #[http://dx.doi.org/10.1070/RM1970v025n02ABEH003794 Ya G Sinai "Dynamical systems with elastic reflections", Russian Mathematical Surveys '''25''' pp. 137-189 (1970)] | ||
#[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/10.1063/1.1446842 Carl McBride and Carlos Vega "Fluid solid equilibrium for two dimensional tangent hard disk chains from Wertheim's perturbation theory", Journal of Chemical Physics '''116''' pp. 1757-1759 (2002)] | #[http://dx.doi.org/10.1063/1.1446842 Carl McBride and Carlos Vega "Fluid solid equilibrium for two dimensional tangent hard disk chains from Wertheim's perturbation theory", Journal of Chemical Physics '''116''' pp. 1757-1759 (2002)] | ||
#[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)] | ||
#[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)] | |||
[[Category: Models]] | [[Category: Models]] |
Revision as of 15:33, 7 August 2008
Hard disks are hard spheres in two dimensions. The hard disk intermolecular pair potential is given by
where is the intermolecular pair potential between two disks at a distance , and is the diameter of the disk.
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. 5) using over 4 million particles one appears to have the phase diagram isotropic hexatic solid.
Equations of state
- Main article: Equations of state for hard disks
Virial coefficients
- Main article: Hard sphere: virial coefficients
External links
- Hard disks and spheres computer code on SMAC-wiki.
References
- 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)
- Carl McBride and Carlos Vega "Fluid solid equilibrium for two dimensional tangent hard disk chains from Wertheim's perturbation theory", Journal of Chemical Physics 116 pp. 1757-1759 (2002)
- Nándor Simányi "Proof of the Boltzmann-Sinai ergodic hypothesis for typical hard disk systems", Inventiones Mathematicae 154 pp. 123-178 (2003)
- C. H. Mak "Large-scale simulations of the two-dimensional melting of hard disks", Physical Review E 73 065104(R) (2006)