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 {\displaystyle X_{i}} taking on values x i {\displaystyle x_{i}} , and some sort of potential function or Hamiltonian H ( x 1 , x 2 , … ) {\displaystyle H(x_{1},x_{2},\dots )} , the partition function is defined as
Z ( β ) = ∑ x i exp ( − β H ( x 1 , x 2 , … ) ) {\displaystyle Z(\beta )=\sum _{x_{i}}\exp \left(-\beta H(x_{1},x_{2},\dots )\right)}
The function H is understood to be a real-valued function on the space of states { X 1 , X 2 , … } {\displaystyle \{X_{1},X_{2},\dots \}} , while β {\displaystyle \beta } is a real-valued free parameter (conventionally, the inverse temperature). The sum over the x i {\displaystyle x_{i}} is understood to be a sum over all possible values that each of the random variables X i {\displaystyle X_{i}} may take. Thus, the sum is to be replaced by an integral when the X i {\displaystyle X_{i}} are continuous, rather than discrete. Thus, one writes
Z ( β ) = ∫ exp ( − β H ( x 1 , x 2 , … ) ) d x 1 d x 2 ⋯ {\displaystyle Z(\beta )=\int \exp \left(-\beta H(x_{1},x_{2},\dots )\right)\,dx_{1}\,dx_{2}\cdots }
for the case of continuously-varying X i {\displaystyle 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 ( β ) = tr ( exp ( − β H ) ) {\displaystyle Z(\beta )=\operatorname {tr} \left(\exp \left(-\beta H\right)\right)}
… excerpt ends here. Continue reading the full article.
