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 sciencetheory]] 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 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]].

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.