It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution.
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.
In particular, it shows how to calculate expectation values and Green's functions, forming a bridge to Fredholm theory.
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.
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 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.
All of these concepts have in common the idea that one value is meant to be kept fixed, as others, interconnected in some complicated way, are allowed to vary.
, even as many different probability distributions can give rise to exactly this same (fixed) value.
To constrain the expectation values in this way, one applies the method of Lagrange multipliers.
For chemistry problems involving chemical reactions, the grand canonical ensemble provides the appropriate foundation, and there are two Lagrange multipliers.
is commonly taken to be real, it need not be, in general; this is discussed in the section Normalization below.
For example, in statistical mechanics, such as the Ising model, the sum is over pairs of nearest neighbors.
Such symmetries can be discrete or continuous; they materialize in the correlation functions for the random variables (discussed below).
This symmetry has a critically important interpretation in probability theory: it implies that the Gibbs measure has the Markov property; that is, it is independent of the random variables in a certain way, or, equivalently, the measure is identical on the equivalence classes of the symmetry.
This leads to the widespread appearance of the partition function in problems with the Markov property, such as Hopfield networks.
Thus, real-valued random fields take values on a simplex: this is the geometrical way of saying that the sum of probabilities must total to one.
For quantum mechanics, the random variables range over complex projective space (or complex-valued projective Hilbert space), where the random variables are interpreted as probability amplitudes.
The partition function is very heavily exploited in the path integral formulation of quantum field theory, to great effect.
The theory there is very nearly identical to that presented here, aside from this difference, and the fact that it is usually formulated on four-dimensional space-time, rather than in a general way.
Given the definition of the probability measure above, the expectation value of any function f of the random variables X may now be written as expected: so, for discrete-valued X, one writes The above notation makes sense for a finite number of discrete random variables.
In more general settings, the summations should be replaced with integrals over a probability space.
Multiple derivatives with regard to the Lagrange multipliers gives rise to a positive semi-definite covariance matrix This matrix is positive semi-definite, and may be interpreted as a metric tensor, specifically, a Riemannian metric.
Equipping the space of lagrange multipliers with a metric in this way turns it into a Riemannian manifold.
That the above defines the Fisher information metric can be readily seen by explicitly substituting for the expectation value: where we've written
Multiple differentiations lead to the connected correlation functions of the random variables.
is given by: For the case where H can be written as a quadratic form involving a differential operator, that is, as then partition function can be understood to be a sum or integral over Gaussians.
can be understood to be the Green's function for the differential operator (and generally giving rise to Fredholm theory).
In the quantum field theory setting, such functions are referred to as propagators; higher order correlators are called n-point functions; working with them defines the effective action of a theory.
Partition functions are used to discuss critical scaling, universality and are subject to the renormalization group.