Lattice hard spheres - Revision history

137.222.75.32: /* Square lattice */

2009-06-22T14:22:38Z

Square lattice

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	== Two-dimensional lattices ==		== Two-dimensional lattices ==
	=== Square lattice ===		=== Square lattice ===
	The model with exclusion of nearest neighbours presents a ~~discontinuous~~ transition. The critical behaviour at the transition		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		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.

Noe: /* Triangular lattice */ less than minor edit

2009-03-27T09:00:13Z

Triangular lattice: less than minor edit

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	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]]
	~~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.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/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>		<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.		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.

Nice and Tidy: Migrated references to Cite.

2009-03-26T13:06:11Z

Migrated references to Cite.

← Older revision		Revision as of 15:06, 26 March 2009
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	These systems exhibit phase (order-disorder) transitions.		These systems exhibit phase (order-disorder) transitions.
	== Three-dimensional lattices ==		== Three-dimensional lattices ==
	~~See Ref. 1 for~~ some results of three-dimensional lattice hard sphere systems (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.		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]] ~~(See Ref 1)~~.		<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 ~~behavior~~ at the transition		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
	~~See Ref 2. 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]] (~~Refs 3~~-5).		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. 3 and Ref.5)~~ 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.		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/>
	~~#[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)]~~
	#[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)]
	#[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)]
	#[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)]
	#[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) (7 pages) ]
	[[category: models]]		[[category: models]]

Noe: /* Square lattice */

2009-03-26T10:23:19Z

Square lattice

← Older revision		Revision as of 12:23, 26 March 2009
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	The model with exclusion of nearest neighbours presents a discontinuous transition. The critical behavior at the transition		The model with exclusion of nearest neighbours presents a discontinuous transition. The critical behavior 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 ~~Revies B~~ 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.

	See Ref 2. for results of two-dimensional systems (lattice hard disks) with different exclusion criteria		See Ref 2. for results of two-dimensional systems (lattice hard disks) with different exclusion criteria

Noe: /* Square lattice */

2009-03-26T10:22:35Z

Square lattice

← Older revision		Revision as of 12:22, 26 March 2009
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	The model with exclusion of nearest neighbours presents a discontinuous transition. The critical behavior at the transition		The model with exclusion of nearest neighbours presents a discontinuous transition. The critical behavior 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'', ~~Phys. Rev.~~ B '''62''', 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 Revies B B '''62''', pp 2134 - 2145 (2000)] </ref> for a simulation study of this system.

	See Ref 2. for results of two-dimensional systems (lattice hard disks) with different exclusion criteria		See Ref 2. for results of two-dimensional systems (lattice hard disks) with different exclusion criteria

Noe: /* Square lattice */

2009-03-26T10:21:13Z

Square lattice

← Older revision		Revision as of 12:21, 26 March 2009
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	The model with exclusion of nearest neighbours presents a discontinuous transition. The critical behavior at the transition		The model with exclusion of nearest neighbours presents a discontinuous transition. The critical behavior 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		<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'', Phys. Rev. B '''62''', 2134 - 2145 (2000)] </ref> for a simulation study of this system.
	''Ordering and percolation transitions for hard squares: Equilibrium versus nonequilibrium models for adsorbed layers with c(2×2) superlattice ordering'', Phys. Rev. B '''62''', 2134 - 2145 (2000)]
	</ref> for a simulation study of this system.

	See Ref 2. for results of two-dimensional systems (lattice hard disks) with different exclusion criteria		See Ref 2. for results of two-dimensional systems (lattice hard disks) with different exclusion criteria

Noe: /* Square lattice */ reference added

2009-03-26T10:20:37Z

Square lattice: reference added

← Older revision		Revision as of 12:20, 26 March 2009
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	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		<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'', Phys. Rev. B '''62''', 2134 - 2145 (2000)		''Ordering and percolation transitions for hard squares: Equilibrium versus nonequilibrium models for adsorbed layers with c(2×2) superlattice ordering'', Phys. Rev. B '''62''', 2134 - 2145 (2000)]
	</ref> for a simulation study of this system.		</ref> for a simulation study of this system.

Noe: /* References */

2009-03-26T10:19:54Z

References

← Older revision		Revision as of 12:19, 26 March 2009
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	== References ==		== References ==
			<references/>
	#[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)]		#[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)]
	#[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)]		#[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)]

Noe: /* Square lattice */

2009-03-26T10:18:14Z

Square lattice

← Older revision		Revision as of 12:18, 26 March 2009
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	=== Square lattice ===		=== Square lattice ===
	The model with exclusion of nearest neighbours presents a discontinuous transition. The critical behavior at the transition		The model with exclusion of nearest neighbours presents a discontinuous transition. The critical behavior at the transition
	corresponds to the same Universality class of the two-dimensional [[Ising model\|Ising Model]].		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'', Phys. Rev. B '''62''', 2134 - 2145 (2000)
			</ref> for a simulation study of this system.

	See Ref 2. for results of two-dimensional systems (lattice hard disks) ~~width~~ different exclusion ~~rules~~		See Ref 2. for results of two-dimensional systems (lattice hard disks) with different exclusion criteria
	on a [[building up a square lattice\|square lattice]].		on a [[building up a square lattice\|square lattice]].

Noe: /* Square lattice */ more details

2009-03-26T10:12:06Z

Square lattice: more details

← Older revision		Revision as of 12:12, 26 March 2009
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	== Two-dimensional lattices ==		== Two-dimensional lattices ==
	=== Square lattice ===		=== Square lattice ===
	See Ref 2. for results of two-dimensional systems (lattice hard disks) on a [[building up a square lattice\|square lattice]].		The model with exclusion of nearest neighbours presents a discontinuous transition. The critical behavior at the transition
			corresponds to the same Universality class of the two-dimensional [[Ising model\|Ising Model]].

			See Ref 2. for results of two-dimensional systems (lattice hard disks) width different exclusion rules
			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]]).