翻訳と辞書 Words near each other ・ Partitio terrarum imperii Romaniae ・ Partition ・ Partition (1987 film) ・ Partition (2007 film) ・ Partition (database) ・ Partition (law) ・ Partition (number theory) ・ Partition (politics) ・ Partition (song) ・ Partition alignment ・ Partition and secession in California ・ Partition cardinal ・ Partition coefficient ・ Partition equilibrium ・ Partition function ・ Partition function (mathematics) ・ Partition function (quantum field theory) ・ Partition function (statistical mechanics) ・ Partition function for Interacting RNAs ・ Partition matroid ・ Partition of a set ・ Partition of Albania ・ Partition of an interval ・ Partition of Babylon ・ Partition of Belgium ・ Partition of Bengal ・ Partition of Bengal (1905) ・ Partition of Bengal (1947) ・ Partition of Bosnia and Herzegovina ・ Partition of Czechoslovakia Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Partition function (mathematics) ： ウィキペディア英語版 Partition function (mathematics) The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution. The partition function occurs in many problems of probability theory because, in situations where there is a natural symmetry, its associated probability measure, the Gibbs measure, has the Markov property. This means that the partition function occurs not only in physical systems with translation symmetry, but also in such varied settings as neural networks (the Hopfield network), and applications such as genomics, corpus linguistics and artificial intelligence, which employ Markov networks, and Markov logic networks. The Gibbs measure is also the unique measure that has the property of maximizing the entropy for a fixed expectation value of the energy; this underlies the appearance of the partition function in maximum entropy methods and the algorithms derived therefrom. The partition function ties together many different concepts, and thus offers a general framework in which many different kinds of quantities may be calculated. In particular, it shows how to calculate expectation values and Green's functions, forming a bridge to Fredholm theory. It also provides a natural setting for the information geometry approach to information theory, where the Fisher information metric can be understood to be a correlation function derived from the partition function; it happens to define a Riemannian manifold. When the setting for random variables is on complex projective space or projective Hilbert space, geometrized with the Fubini–Study metric, the theory of quantum mechanics and more generally quantum field theory results. In these theories, the partition function is heavily exploited in the path integral formulation, with great success, leading to many formulas nearly identical to those reviewed here. However, because the underlying measure space is complex-valued, as opposed to the real-valued simplex of probability theory, an extra factor of ''i'' appears in many formulas. Tracking this factor is troublesome, and is not done here. This article focuses primarily on classical probability theory, where the sum of probabilities total to one. ==Definition== Given a set of random variables $X\_i$ taking on values $x\_i$, and some sort of potential function or Hamiltonian $H(x\_1,x\_2,\backslash dots)$, the partition function is defined as :$Z(\backslash beta)\; =\; \backslash sum\_\; \backslash exp\; \backslash left(-\backslash beta\; H(x\_1,x\_2,\backslash dots)\; \backslash right)$ The function ''H'' is understood to be a real-valued function on the space of states $\backslash$, while $\backslash beta$ is a real-valued free parameter (conventionally, the inverse temperature). The sum over the $x\_i$ is understood to be a sum over all possible values that each of the random variables $X\_i$ may take. Thus, the sum is to be replaced by an integral when the $X\_i$ are continuous, rather than discrete. Thus, one writes :$Z(\backslash beta)\; =\; \backslash int\; \backslash exp\; \backslash left(-\backslash beta\; H(x\_1,x\_2,\backslash dots)\; \backslash right)\; dx\_1\; dx\_2\; \backslash cdots$ for the case of continuously-varying $X\_i$. When ''H'' is an observable, such as a finite-dimensional matrix or an infinite-dimensional Hilbert space operator or element of a C-star algebra, it is common to express the summation as a trace, so that :$Z(\backslash beta)\; =\; \backslash mbox\backslash left(\backslash exp\backslash left(-\backslash beta\; H\backslash right)\backslash right)$ When ''H'' is infinite-dimensional, then, for the above notation to be valid, the argument must be trace class, that is, of a form such that the summation exists and is bounded. The number of variables $X\_i$ need not be countable, in which case the sums are to be replaced by functional integrals. Although there are many notations for functional integrals, a common one would be :$Z\; =\; \backslash int\; \backslash mathcal\; \backslash phi\; \backslash exp\; \backslash left(-\; \backslash beta\; H()\; \backslash right)$ Such is the case for the partition function in quantum field theory. A common, useful modification to the partition function is to introduce auxiliary functions. This allows, for example, the partition function to be used as a generating function for correlation functions. This is discussed in greater detail below. 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「Partition function (mathematics)」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.