Lattice hard spheres: Difference between revisions
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'''Lattice hard spheres''' refers to athermal [[lattice gas|lattice gas]] models, in which pairs | '''Lattice hard spheres''' (or '''Lattice hard disks''') refers to athermal [[lattice gas|lattice gas]] models, in which pairs of sites separated by less than some (short) distance, <math> \sigma </math>, cannot be simultaneously occupied. | ||
of sites separated by less than some short distance <math> \sigma </math> cannot be simultaneously occupied. | |||
== Brief description of the models == | == Brief description of the models == | ||
Basically the differences | Basically the differences between lattice hard spheres and the standard [[Lattice gas|lattice gas]] model ([[Ising Models|Ising model]]) are the following: | ||
*An occupied site excludes the occupation of some of the neighbouring sites. | |||
*An occupied site excludes the occupation of some of the | *No energy interactions between pairs of occupied sites -apart of the hard core interactions- are considered. | ||
These systems exhibit phase (order-disorder) transitions. | |||
*No energy interactions between pairs of occupied sites are considered. | |||
These systems exhibit phase (order-disorder) transitions | |||
== Three-dimensional lattices == | == Three-dimensional lattices == | ||
For some results of three-dimensional lattice hard sphere systems see | |||
<ref>[http://dx.doi.org/10.1063/1.2008253 A. Z. Panagiotopoulos, "Thermodynamic properties of lattice hard-sphere models", Journal of Chemical Physics '''123''' 104504 (2005)]</ref> (on a [[Building up a simple cubic lattice |simple cubic lattice]]). The model defined on a simple cubic lattice with exclusion of only the nearest neighbour positions of an occupied site presents a continuous transition. | |||
If next-nearest neighbours are also excluded then the transition becomes [[First-order transitions |first order]]. | |||
== Two-dimensional lattices == | == Two-dimensional lattices == | ||
=== Square lattice === | === Square lattice === | ||
The model with exclusion of nearest neighbours presents a continuous transition. The critical behaviour at the transition | |||
corresponds to the same Universality class of the two-dimensional [[Ising model|Ising Model]], See Ref | |||
<ref>[http://dx.doi.org/10.1103/PhysRevB.62.2134 Da-Jiang Liu and J. W. Evans, "Ordering and percolation transitions for hard squares: Equilibrium versus nonequilibrium models for adsorbed layers with c(2×2) superlattice ordering", Physical Review B '''62''', pp 2134 - 2145 (2000)] </ref> for a simulation study of this system. | |||
For results of two-dimensional systems (lattice hard disks) with different exclusion criteria | |||
on a [[building up a square lattice|square lattice]] see <ref>[http://dx.doi.org/10.1063/1.2539141 Heitor C. Marques Fernandes, Jeferson J. Arenzon, and Yan Levin "Monte Carlo simulations of two-dimensional hard core lattice gases", Journal of Chemical Physics '''126''' 114508 (2007)]</ref>. | |||
=== Triangular lattice === | === [[Building up a triangular lattice|Triangular lattice]] === | ||
The [[hard hexagon lattice model|hard hexagon lattice model]] belongs to this kind of | The [[hard hexagon lattice model|hard hexagon lattice model]] belongs to this kind of model. In this model an occupied site excluded the occupation of nearest neighbour positions. This model exhibits a continuous transition, and has been solved exactly (See references in the entry: [[hard hexagon lattice model|hard hexagon lattice model]]). | ||
excluded the occupation of nearest neighbour positions. This model exhibits a | Other models defined on the triangular lattice (with more excluded positions) have been studied theoretically and by [[Monte Carlo | Monte Carlo simulation]] | ||
in the entry: [[hard hexagon lattice model|hard hexagon lattice model]]). | <ref>[http://dx.doi.org/10.1103/PhysRevB.30.5339 N. C. Bartelt and T. L. Einstein, "Triangular lattice gas with first- and second-neighbor exclusions: Continuous transition in the four-state Potts universality class", Physical Review B '''30''' pp. 5339-5341 (1984)]</ref> | ||
<ref>[http://dx.doi.org/10.1103/PhysRevB.39.2948 Chin-Kun Hu and Kit-Sing Mak, "Percolation and phase transitions of hard-core particles on lattices: Monte Carlo approach", Physical Review B '''39''' pp. 2948-2951 (1989)]</ref> | |||
Other models defined on the triangular lattice (with more excluded positions) have been studied theoretically and by [[Monte Carlo | Monte Carlo simulation]]. | <ref>[http://dx.doi.org/10.1103/PhysRevE.78.031103 Wei Zhang Youjin Den, ''Monte Carlo study of the triangular lattice gas with first- and second-neighbor exclusions'', Physical Review E '''78''' 031103 (2008)]</ref>. | ||
It seems that the model with first and second neighbour exclusion presents also a continuous transition, whereas if third neighbours are also excluded the transition becomes first order. | |||
== References == | == References == | ||
<references/> | |||
[[category: models]] | [[category: models]] | ||
Latest revision as of 15:22, 22 June 2009
Lattice hard spheres (or Lattice hard disks) refers to athermal lattice gas models, in which pairs of sites separated by less than some (short) distance, , cannot be simultaneously occupied.
Brief description of the models[edit]
Basically the differences between lattice hard spheres and the standard lattice gas model (Ising model) are the following:
- An occupied site excludes the occupation of some of the neighbouring sites.
- No energy interactions between pairs of occupied sites -apart of the hard core interactions- are considered.
These systems exhibit phase (order-disorder) transitions.
Three-dimensional lattices[edit]
For some results of three-dimensional lattice hard sphere systems see [1] (on a simple cubic lattice). The model defined on a simple cubic lattice with exclusion of only the nearest neighbour positions of an occupied site presents a continuous transition. If next-nearest neighbours are also excluded then the transition becomes first order.
Two-dimensional lattices[edit]
Square lattice[edit]
The model with exclusion of nearest neighbours presents a continuous transition. The critical behaviour at the transition corresponds to the same Universality class of the two-dimensional Ising Model, See Ref [2] for a simulation study of this system. For results of two-dimensional systems (lattice hard disks) with different exclusion criteria on a square lattice see [3].
Triangular lattice[edit]
The hard hexagon lattice model belongs to this kind of model. In this model an occupied site excluded the occupation of nearest neighbour positions. This model exhibits a continuous transition, and has been solved exactly (See references in the entry: hard hexagon lattice model). Other models defined on the triangular lattice (with more excluded positions) have been studied theoretically and by Monte Carlo simulation [4] [5] [6]. It seems that the model with first and second neighbour exclusion presents also a continuous transition, whereas if third neighbours are also excluded the transition becomes first order.
References[edit]
- ↑ A. Z. Panagiotopoulos, "Thermodynamic properties of lattice hard-sphere models", Journal of Chemical Physics 123 104504 (2005)
- ↑ Da-Jiang Liu and J. W. Evans, "Ordering and percolation transitions for hard squares: Equilibrium versus nonequilibrium models for adsorbed layers with c(2×2) superlattice ordering", Physical Review B 62, pp 2134 - 2145 (2000)
- ↑ Heitor C. Marques Fernandes, Jeferson J. Arenzon, and Yan Levin "Monte Carlo simulations of two-dimensional hard core lattice gases", Journal of Chemical Physics 126 114508 (2007)
- ↑ N. C. Bartelt and T. L. Einstein, "Triangular lattice gas with first- and second-neighbor exclusions: Continuous transition in the four-state Potts universality class", Physical Review B 30 pp. 5339-5341 (1984)
- ↑ Chin-Kun Hu and Kit-Sing Mak, "Percolation and phase transitions of hard-core particles on lattices: Monte Carlo approach", Physical Review B 39 pp. 2948-2951 (1989)
- ↑ Wei Zhang Youjin Den, Monte Carlo study of the triangular lattice gas with first- and second-neighbor exclusions, Physical Review E 78 031103 (2008)