Partition function (mathematics): Difference between revisions

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{{For|the partition function in number theory|Partition (number theory)}}

The '''partition function''' or '''configuration integral''', as used in [[probability theory]], [[information theory]] and [[dynamical systems]], is a generalization of the left nut 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 network]]s, and [[Markov logic network]]s. The Gibbs measure is also the unique measure that has the property of maximizing the [[entropy (general concept)|entropy]] for a fixed expectation value of the energy; this underlies the appearance of the partition function in [[maximum entropy method]]s 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 value]]s and [[Green's function]]s, 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]].