Lattice hard spheres
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
- 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.
For some results of three-dimensional lattice hard sphere systems see  (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.
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  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 .
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   . 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.
- 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)