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In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems. The canonical ensemble gives the probability of the system ''X'' being in state ''x'' (equivalently, of the random variable ''X'' having value ''x'') as : Here, is a function from the space of states to the real numbers; in physics applications, is interpreted as the energy of the configuration ''x''. The parameter is a free parameter; in physics, it is the inverse temperature. The normalizing constant is the partition function. However, in infinite systems, the total energy is no longer a finite number and cannot be used in the traditional construction of the probability distribution of a canonical ensemble. Traditional approaches in statistical physics studied the limit of intensive properties as the size of a finite system approaches infinity (the thermodynamic limit). When the energy function can be written as a sum of terms that each involve only variables from a finite subsystem, the notion of a Gibbs measure provides an alternative approach. Gibbs measures were proposed by probability theorists such as Dobrushin, Lanford, and Ruelle and provided a framework to directly study infinite systems, instead of taking the limit of finite systems. A measure is a Gibbs measure if the conditional probabilities it induces on each finite subsystem satisfy a consistency condition: if all degrees of freedom outside the finite subsystem are frozen, the canonical ensemble for the subsystem subject to these boundary conditions matches the probabilities in the Gibbs measure conditional on the frozen degrees of freedom. The Hammersley–Clifford theorem implies that any probability measure that satisfies a Markov property is a Gibbs measure for an appropriate choice of (locally defined) energy function. Therefore, the Gibbs measure applies to widespread problems outside of physics, such as Hopfield networks, Markov networks, and Markov logic networks. A Gibbs measure in a system with local (finite-range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density. The Gibbs measure of an infinite system is not necessarily unique, in contrast to the canonical ensemble of a finite system, which is unique. The existence of more than one Gibbs measures is associated with statistical phenomena such as symmetry breaking and phase coexistence. ==Markov property== An example of the Markov property can be seen in the Gibbs measure of the Ising model. The probability for a given spin to be in state ''s'' could, in principle, depend on the states of all other spins in the system. Thus, we may write the probability as :. However, in an Ising model with only finite-range interactions (for example, nearest-neighbor interactions), we actually have :, where is a neighborhood of the site . That is, the probability at site depends ''only'' on the spins in a finite neighborhood. This last equation is in the form of a local Markov property. Measures with this property are sometimes called Markov random fields. More strongly, the converse is also true: ''any'' positive probability distribution (nonzero density everywhere) having the Markov property can be represented as a Gibbs measure for an appropriate energy function.〔Ross Kindermann and J. Laurie Snell, (Markov Random Fields and Their Applications ) (1980) American Mathematical Society, ISBN 0-8218-5001-6〕 This is the Hammersley–Clifford theorem. 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「Gibbs measure」の詳細全文を読む スポンサード リンク
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