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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 model was solved by , who found that it was related to the Rogers–Ramanujan identities. ==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 ''z''''c'' = (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 : : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「hard hexagon model」の詳細全文を読む スポンサード リンク
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