Lattice hard spheres: Difference between revisions
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These systems exhibit phase (order-disorder) transitions. | These systems exhibit phase (order-disorder) transitions. | ||
== Three-dimensional lattices == | == Three-dimensional lattices == | ||
For some results of three-dimensional lattice hard sphere systems see | |||
If next-nearest neighbours are also excluded then the transition becomes [[First-order transitions |first order]] | <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 discontinuous transition. The critical | The model with exclusion of nearest neighbours presents a discontinuous transition. The critical behaviour at the transition | ||
corresponds to the same Universality class of the two-dimensional [[Ising model|Ising Model]], See Ref | 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. | <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>. | |||
on a [[building up a square lattice|square lattice]]. | |||
=== [[Building up a triangular lattice|Triangular lattice]] === | === [[Building up a triangular lattice|Triangular lattice]] === | ||
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]]). | 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]]). | ||
Other models defined on the triangular lattice (with more excluded positions) have been studied theoretically and by [[Monte Carlo | Monte Carlo simulation]] ( | Other models defined on the triangular lattice (with more excluded positions) have been studied theoretically and by [[Monte Carlo | Monte Carlo simulation]] | ||
It seems | see | ||
<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> | |||
<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/> | <references/> | ||
[[category: models]] | [[category: models]] | ||
Revision as of 14:06, 26 March 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, 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 } , cannot be simultaneously occupied.
Brief description of the models
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
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
Square lattice
The model with exclusion of nearest neighbours presents a discontinuous 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
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 see [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
- ↑ 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)