- Partition function (mathematics)
The partition function or configuration integral, as used in
probability theory, information scienceand dynamical systems, is an abstraction of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constantin 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 linguisticsand 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.
Given a set of
random variablestaking on values , and some sort of of potential functionor Hamiltonian , the partition function is defined as
The function "H" is understood to be a real-valued function on the space of states , while is a real-valued free parameter (conventionally, the
inverse temperature). The sum over the is understood to be a sum over all possible values that the random variable may take. Thus, the sum is to be replaced by an integralwhen the are continuous, rather than discrete. Thus, one writes
for the case of continuously-varying .
The number of variables need not be
countable, in which case the sums are to be replaced by functional integrals. Although there are many notations for functional integrals, a common one would be
Such is the case for for the
partition function in quantum field theory.
A common, useful modification to the partition function is to introduce auxiliary functions. This allows, for example, the partition function to be used as a
generating functionfor correlation functions. This is discussed in greater detail below.
The parameter β
The role or meaning of the parameter is best understood by examining the derivation of the partition function with
maximum entropy methods. Here, the parameter appears as a Lagrange multiplier; the multiplier is used to guarantee that the expectation valueof some quantity is preserved by the distribution of probabilities. Thus, in physics problems, the use of just one parameter reflects the fact that there is only one expectation value that must be held constant: this is the energy. For the grand canonical ensemble, there are two Lagrange multipliers: one to hold the energy constant, and another (the fugacity) to hold the particle count constant. In the general case, there are a set of parameters taking the place of , one for each constraint enforced by the multiplier. Thus, for the general case, one has
Gibbs measurethen provides a probability distribution such that the expectation value of each is a fixed value.
Although the value of is commonly taken to be real, it need not be, in general; this is discussed in the section Normalization below.
The potential function itself commonly takes the form of a sum:
where the sum over "s" is a sum over some subset of the
power set"P"("X") of the set . For example, in statistical mechanics, such as the Ising model, the sum is over pairs of nearest neighbors. In probability theory, such as Markov networks, the sum might be over the cliques of a graph; so, for the Ising model and other lattice models, the maximal cliques are edges.
The fact that the potential function can be written as a sum usually reflects the fact that it is invariant under the action of a group symmetry, such as
translational invariance. Such symmetries can be discrete or continuous; they materialize in the correlation functions for the random variables (discussed below). Thus a symmetry in the Hamiltonian becomes a symmetry of the correlation function (and vice-versa).
This symmetry has a critically important interpretation in probability theory: it implies that the
Gibbs measurehas 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.
As a measure
The value of the expression :
can be interpreted as a likelihood that a specific configuration of values occurs in the system. Thus, given a specific configuration ,
is the probability of the configuration occurring in the system, which is now properly normalized so that , and such that the sum over all configurations totals to one. As such, the partition function can be understood to provide a measure on the space of states; it is sometimes called the
Gibbs measure. More narrowly, it is called the canonical ensemblein statistical mechanics.
There exists at least one configuration for which the probability is maximized; this configuration is conventionally called the
ground state. If the configuration is unique, the ground state is said to be non-degenerate, and the system is said to be ergodic; otherwise the ground state is degenerate. The ground state may or may not commute with the generators of the symetry; if commutes, it is said to be an invariant measure. When it does not commute, the symmetry is said to be spontaneously broken.
Conditions under which a ground state exists and is unique are given by the
Karush–Kuhn–Tucker conditions; these conditions are commonly used to justify the use of the Gibbs measure in maximum-entropy problems.
The values taken by depend on the
mathematical spaceover which the random field varies. Thus, real-valued random fields take values on a simplex: this the geometrical way of saying that the sum of probabilities must total to one. For quantum mechanics, the random variables ranges over complex projective space(or complex-valued Hilbert space), because the random variables are interpreted as probability amplitudes. The emphasis here is on the word "projective", as the amplitudes are still normalized to one. The normalization for the potential function is the Jacobianfor the appropriate mathematical space: it is 1 for ordinary probabilities, and "i" for complex Hilbert space; thus, in quantum field theory, one sees "iS" in the exponential, rather than .
The partition function is commonly used as a
generating functionfor expectation values of various functions of the random variables. So, for example, taking as an adjustable parameter, then the derivative of with respect to
gives the average (expectation value) of "H". In physics, this would be called the average
energyof the system.
The entropy is given by
The Gibbs measure is the unique statistical distribution that maximizes the entropy for a fixed expectation value of the energy; this underlies its use in
maximum entropy methods.
By introducing artificial auxiliary functions into the partition function, it can then be used to obtain the expectation value of the random variables. Thus, for example, by writing
one then has :
as the expectation value of .
Multiple differentiations lead to the
correlation functions of the random variables. Thus the correlation function between variables and is given by:
For the case where "H" can be written as a
quadratic forminvolving a differential operator, that is, as
then the correlation function can be understood to be the
Green's functionfor the differential operator (and generally giving rise to Fredholm theory).
Partition functions often show
critical scaling, universality and are subject to the renormalization group.
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