Hard hexagon model
In statistical mechanics, the hard hexagon model is a 2-dimensional lattice model of a gas, where particles are allowed to be on the vertices of a triangular lattice but no two particles may be adjacent.
The partition function of the hard hexagon model
For a triangular lattice with N sites, the grand partition function is
where g(n, N) is the number of ways of placing n particles on distinct lattice sites such that no 2 are adjacent. The variable z is called the activity and larger values correspond roughly to denser configurations. The function κ is defined by
so that log(κ) is the free energy per unit site. Solving the hard hexagon model means (roughly) finding an exact expression for κ as a function of z.
The mean density ρ is given for small z by
The vertices of the lattice fall into 3 classes numbered 1, 2, and 3, given by the 3 different ways to fill space with hard hexagons. There are 3 local densities ρ1, ρ2, ρ3, corresponding to the 3 classes of sites. When the activity is large the system approximates one of these 3 packings, so the local densities differ, but when the activity is below a critical point the three local densities are the same. The critical point separating the low-activity homogeneous phase from the high-activity ordered phase is zc = (11 + 53/2)/2 = 11.0917.... Above the critical point the local densities differ and in the phase where most hexagons are on sites of type 1 can be expanded as
The solution is given for small values of z < zc by
For large z > zc the solution (in the phase where most occupied sites have type 1) is given by
The functions G and H turn up in the Rogers–Ramanujan identities, and the function Q is closely related to the Dedekind eta function. If x = e2πiτ, then q−1/60G(x), x11/60H(x), x−1/24P(x), z, κ, ρ, ρ1, ρ2, and ρ3 are modular functions of τ, while x1/24Q(x) is a modular form of weight 1/2. Since any two modular functions are related by an algebraic relation, this implies that the functions κ, z, R, ρ are all algebraic functions of each other (of quite high degree) (Joyce 1988).
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- Joyce, G. S. (1988), "Exact results for the activity and isothermal compressibility of the hard-hexagon model", Journal of Physics. A. Mathematical and General 21 (20): L983–L988, doi:10.1088/0305-4470/21/20/005, ISSN 0305-4470, MR 966792