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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 Ice-type model ： ウィキペディア英語版 Ice-type model In statistical mechanics, the ice-type models or six-vertex models are a family of vertex models for crystal lattices with hydrogen bonds. The first such model was introduced by Linus Pauling in 1935 to account for the residual entropy of water ice.〔 〕 Variants have been proposed as models of certain ferroelectric〔 〕 and antiferroelectric〔 〕 crystals. In 1967, Elliott H. Lieb found the exact solution to a two-dimensional ice model known as "square ice".〔 〕 The exact solution in three dimensions is only known for a special "frozen" state.〔 〕 ==Description== An ice-type model is a lattice model defined on a lattice of coordination number 4 - that is, each vertex of the lattice is connected by an edge to four "nearest neighbours". A state of the model consists of an arrow on each edge of the lattice, such that the number of arrows pointing inwards at each vertex is 2. This restriction on the arrow configurations is known as the ice rule. In graph theoretic terms, the states are Eulerian orientations of the underlying undirected graph.〔 〕 For two-dimensional models, the lattice is taken to be the square lattice. For more realistic models, one can use a three-dimensional lattice appropriate to the material being considered; for example, the hexagonal ice lattice is used to analyse ice. At any vertex, there are six configurations of the arrows which satisfy the ice rule (justifying the name "six-vertex model"). The valid configurations for the (two-dimensional) square lattice are the following: :500px The energy of a state is understood to be a function of the configurations at each vertex. For square lattices, one assumes that the total energy $E$ is given by :$E\; =\; n\_1\backslash epsilon\_1\; +\; n\_2\backslash epsilon\_2\; +\; \backslash ldots\; +\; n\_6\backslash epsilon\_6,$ for some constants $\backslash epsilon\_1,\backslash ldots,\backslash epsilon\_6$, where $n\_i$ here denotes the number of vertices with the $i$th configuration from the above figure. The value $\backslash epsilon\_i$ is the energy associated with vertex configuration number $i$. One aims to calculate the partition function $Z$ of an ice-type model, which is given by the formula :$Z\; =\; \backslash sum\; \backslash exp(-E/k\_BT),$ where the sum is taken over all states of the model, $E$ is the energy of the state, $k\_B$ is Boltzmann's constant, and $T$ is the system's temperature. Typically, one is interested in the thermodynamic limit in which the number $N$ of vertices approaches infinity. In that case, one instead evaluates the free energy per vertex $f$ in the limit as $N\backslash to\; \backslash infty$, where $f$ is given by :$f\; =\; -k\_BT\; N^\backslash log\; Z.$ Equivalently, one evaluates the partition function per vertex $W$ in the thermodynamic limit, where :$W=Z^.$ The values $f$ and $W$ are related by :$f=\; -k\_BT\; \backslash log\; W.$ 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「Ice-type 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.