Editing Lattice hard spheres

'''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.
== Brief description of the models ==
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.
*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 ==
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.
If next-nearest neighbours are also excluded then the transition becomes [[First-order transitions |first order]] (See Ref 1).
== Two-dimensional lattices ==
=== 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 
<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) with different exclusion criteria
on a [[building up a square lattice|square 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]]).
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).
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.

== 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]]