Lattice hard spheres

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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[edit]

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[edit]

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[edit]

Square lattice[edit]

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 [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[edit]

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 [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[edit]