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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 vertex model ： ウィキペディア英語版 vertex model A vertex model is a type of statistical mechanics model in which the Boltzmann weights are associated with a vertex in the model (representing an atom or particle).〔R.J. Baxter, ''Exactly solved models in statistical mechanics'', London, Academic Press, 1982〕〔V. Chari and A.N. Pressley, ''A Guide to Quantum Groups'' Cambridge University Press, 1994〕 This contrasts with a nearest-neighbour model, such as the Ising model, in which the energy, and thus the Boltzmann weight of a statistical microstate is attributed to the bonds connecting two neighbouring particles. The energy associated with a vertex in the lattice of particles is thus dependent on the state of the bonds which connect it to adjacent vertices. It turns out that every solution of the Yang-Baxter equation with spectral parameters in a tensor product of vector spaces $V\backslash otimes\; V$ yields an exactly-solvable vertex model. Although the model can be applied to various geometries in any number of dimensions, with any number of possible states for a given bond, the most fundamental examples occur for two dimensional lattices, the simplest being a square lattice where each bond has two possible states. In this model, every particle is connected to four other particles, and each of the four bonds adjacent to the particle has two possible states, indicated by the direction of an arrow on the bond. In this model, each vertex can adopt $2^4$ possible configurations. The energy for a given vertex can be given by $\backslash varepsilon\_^$, with a state of the lattice is an assignment of a state of each bond, with the total energy of the state being the sum of the vertex energies. As the energy is often divergent for an infinite lattice, the model is studied for a finite lattice as the lattice approaches infinite size. Periodic or domain wall〔V.E. Korepin et al., ''Quantum inverse scattering method and correlation functions'', New York, Press Syndicate of the University of Cambridge, 1993〕 boundary conditions may be imposed on the model. ==Discussion== For a given state of the lattice, the Boltzmann weight can be written as the product over the vertices of the Boltzmann weights of the corresponding vertex states :$\backslash exp(-\backslash beta\; \backslash varepsilon(\backslash mbox))\; =\; \backslash prod\_\backslash mbox\; \backslash exp(-\backslash beta\; \backslash varepsilon\_^)$ where the Boltzmann weights for the vertices are written :$R\_^\; =\; \backslash exp(-\backslash beta\; \backslash varepsilon\_^)$, and the ''i'', ''j'', ''k'', ''l'' range over the possible statuses of each of the four edges attached to the vertex. The vertex states of adjacent vertices must satisfy compatibility conditions along the connecting edges (bonds) in order for the state to be admissible. The probability of the system being in any given state at a particular time, and hence the properties of the system are determined by the partition function, for which an analytic form is desired. :$\backslash mathbb\; =\; \backslash sum\_\backslash mbox\; \backslash exp(-\backslash beta\; \backslash varepsilon(\backslash mbox))$ where β=''1/kT'', ''T'' is temperature and ''k'' is Boltzmann's constant. The probability that the system is in any given state (microstate) is given by :$\backslash frac\; \backslash varepsilon\; \backslash exp(-\backslash beta\; \backslash varepsilon)\}$ = kT^2 \frac \ln \mathbb In order to evaluate the partition function, firstly examine the states of a row of vertices. The external edges are free variables, with summation over the internal bonds. Hence, form the row partition function :$T\_^\; =\; \backslash sum\_^\; R\_^\; \backslash cdots\; R\_^$ This can be reformulated in terms of an auxiliary ''n''-dimensional vector space ''V'', with a basis $\backslash$, and $R\; \backslash in\; End(V\; \backslash otimes\; V)$ as :$R(v\_i\; \backslash otimes\; v\_j)\; =\; \backslash sum\_\; R\_^\; v\_k\; \backslash otimes\; v\_\backslash ell$ and $T\; \backslash in\; End(V\; \backslash otimes\; V^)$ as :$T(v\_\; \backslash otimes\; v\_\; \backslash otimes\; \backslash cdots\; \backslash otimes\; v\_)\; =\; \backslash sum\_\; T\_^\; v\_\; \backslash otimes\; v\_\; \backslash otimes\; \backslash cdots\; \backslash otimes\; v\_$ thereby implying that ''T'' can be written as :$T\; =\; R\_R\_\backslash cdots\; R\_$ where the indices indicate the factors of the tensor product $V\; \backslash otimes\; V^$ on which ''R'' operates. Summing over the states of the bonds in the first row with the periodic boundary conditions $i\_1\; =\; i\text{'}\_1$, gives :$(\backslash operatorname\_(T))\_^$ where $\backslash tau\; =\; \backslash operatorname\_(T)$ is the row-transfer matrix. By summing the contributions over two rows, the result is :$(\backslash operatorname\_(T))\_^\; (\backslash operatorname\_(T))\_^$ which upon summation over the vertical bonds connecting the first two rows gives:$((\backslash operatorname\_(T))^2)\_^$ for ''M'' rows, this gives :$((\backslash operatorname\_(T))^M)\_^$ and then applying the periodic boundary conditions to the vertical columns, the partition function can be expressed in terms of the transfer matrix $\backslash tau$ as :$\backslash mathbb=\; \backslash operatorname\_^M$ where $\backslash lambda\_$ is the largest eigenvalue of $\backslash tau$. The approximation follows from the fact that the eigenvalues of $\backslash tau^M$ are the eigenvalues of $\backslash tau$ to the power of ''M'', and as $M\; \backslash rightarrow\; \backslash infty$, the power of the largest eigenvalue becomes much larger than the others. As the trace is the sum of the eigenvalues, the problem of calculating $\backslash mathbb$ reduces to the problem of finding the maximum eigenvalue of $\backslash tau$. This in it itself is another field of study. However, a standard approach to the problem of finding the largest eigenvalue of $\backslash tau$ is to find a large family of operators which commute with $\backslash tau$. This implies that the eigenspaces are common, and restricts the possible space of solutions. Such a family of commuting operators is usually found by means of the Yang-Baxter equation, which thus relates statistical mechanics to the study of quantum groups. 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「vertex 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.