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, $\sigma$, 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

See Ref. 1 for some results of three-dimensional lattice hard sphere systems (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 (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, See 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 square lattice.

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 (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

1. A. Z. Panagiotopoulos, "Thermodynamic properties of lattice hard-sphere models", Journal of Chemical Physics 123 104504 (2005)
2. 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)
3. 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)
4. 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)
5. 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)