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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hard hexagon model ： ウィキペディア英語版 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 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 :$\backslash displaystyle\; \backslash mathcal\; Z(z)\; =\; \backslash sum\_n\; z^n\; g(n,N)\; =\; 1+Nz+\; \backslash tfracN(N-7)z^2+\backslash cdots$ 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 :$\backslash kappa(z)\; =\; \backslash lim\_\; \backslash mathcal\; Z(z)^\; =\; 1+z-3z^2+\backslash cdots$ 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 :$\backslash rho=\; z\backslash frac\; =z-7z^2+58z^3-519z^4+4856z^5+\backslash cdots.$ 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 :$\backslash rho\_1\; =\; 1-z^-5z^-34z^-267z^-2037z^-\backslash cdots$ :$\backslash rho\_2=\backslash rho\_3\; =\; z^\; +\; 9z^\; +\; 80z^\; +\; 965z^-\backslash cdots.$ 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 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.