- Partition function (mathematics)
The

**partition function**or**configuration integral**, as used inprobability theory ,information science anddynamical systems , is an abstraction of the definition of apartition function in statistical mechanics . It is a special case of anormalizing constant in probability theory, for theBoltzmann distribution . The partition function occurs in many problems of probability theory because, in situations where there is a natural symmetry, its associatedprobability measure , theGibbs measure , has theMarkov 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 (theHopfield network ), and applications such asgenomics ,corpus linguistics andartificial intelligence , which employMarkov network s, andMarkov logic network s. 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 inmaximum entropy method s and the algorithms derived therefrom.**Definition**Given a set of

random variables $X\_i$ taking on values $x\_i$, and some sort of ofpotential function or Hamiltonian $H(x\_1,x\_2,dots)$, the partition function is defined as:$Z(eta)\; =\; sum\_\{x\_i\}\; exp\; left(-eta\; H(x\_1,x\_2,dots)\; ight)$

The function "H" is understood to be a real-valued function on the space of states $\{X\_1,X\_2,cdots\}$, while $eta$ is a real-valued free parameter (conventionally, the

inverse temperature ). The sum over the $x\_i$ is understood to be a sum over all possible values that the random variable $X\_i$ may take. Thus, the sum is to be replaced by anintegral when the $X\_i$ are continuous, rather than discrete. Thus, one writes:$Z(eta)\; =\; int\; exp\; left(-eta\; H(x\_1,x\_2,dots)\; ight)\; dx\_1\; dx\_2\; cdots$

for the case of continuously-varying $X\_i$.

The number of variables $X\_i$ need not be

countable , in which case the sums are to be replaced byfunctional integral s. Although there are many notations for functional integrals, a common one would be:$Z\; =\; int\; mathcal\{D\}\; phi\; exp\; left(-\; H\; [phi]\; ight)$

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 function forcorrelation function s. This is discussed in greater detail below.**The parameter β**The role or meaning of the parameter $eta$ is best understood by examining the derivation of the partition function with

maximum entropy method s. Here, the parameter appears as aLagrange multiplier ; the multiplier is used to guarantee that theexpectation value of some quantity is preserved by the distribution of probabilities. Thus, in physics problems, the use of just one parameter $eta$ reflects the fact that there is only one expectation value that must be held constant: this is the energy. For thegrand canonical ensemble , there are two Lagrange multipliers: one to hold the energy constant, and another (thefugacity ) to hold the particle count constant. In the general case, there are a set of parameters taking the place of $eta$, one for each constraint enforced by the multiplier. Thus, for the general case, one has:$Z(eta\_k)\; =\; sum\_\{x\_i\}\; exp\; left(-sum\_keta\_k\; H\_k(x\_i)\; ight)$

The corresponding

Gibbs measure then provides a probability distribution such that the expectation value of each $H\_k$ is a fixed value.Although the value of $eta$ is commonly taken to be real, it need not be, in general; this is discussed in the section Normalization below.

**Symmetry**The potential function itself commonly takes the form of a sum:

:$H(x\_1,x\_2,dots)\; =\; sum\_s\; E(s),$

where the sum over "s" is a sum over some subset of the

power set "P"("X") of the set $X=lbrace\; x\_1,x\_2,dots\; brace$. For example, instatistical mechanics , such as theIsing model , the sum is over pairs of nearest neighbors. In probability theory, such asMarkov 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 thecorrelation function s 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 measure has theMarkov property ; that is, it is independent of the random variables in a certain way, or, equivalently, the measure is identical on theequivalence class es of the symmetry. This leads to the widespread appearance of the partition function in problems with the Markov property, such asHopfield network s.**As a measure**The value of the expression :$exp\; left(-eta\; H(x\_1,x\_2,dots)\; ight)$

can be interpreted as a likelihood that a specific configuration of values $(x\_1,x\_2,dots)$ occurs in the system. Thus, given a specific configuration $(x\_1,x\_2,dots)$,

:$P(x\_1,x\_2,dots)\; =\; frac\{1\}\{Z(eta)\}\; exp\; left(-eta\; H(x\_1,x\_2,dots)\; ight)$

is the probability of the configuration $(x\_1,x\_2,dots)$ occurring in the system, which is now properly normalized so that $0le\; P(x\_1,x\_2,dots)le\; 1$, 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 thecanonical ensemble in statistical mechanics.There exists at least one configuration $(x\_1,x\_2,dots)$ 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 beergodic ; 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 aninvariant measure . When it does not commute, the symmetry is said to bespontaneously 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.**Normalization**The values taken by $eta$ depend on the

mathematical space over which the random field varies. Thus, real-valued random fields take values on asimplex : this the geometrical way of saying that the sum of probabilities must total to one. For quantum mechanics, the random variables ranges overcomplex projective space (or complex-valuedHilbert space ), because the random variables are interpreted asprobability amplitude s. The emphasis here is on the word "projective", as the amplitudes are still normalized to one. The normalization for the potential function is theJacobian for 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 $eta\; H$.**Expectation values**The partition function is commonly used as a

generating function forexpectation value s of various functions of the random variables. So, for example, taking $eta$ as an adjustable parameter, then the derivative of $Z(eta)$ with respect to $eta$:$old\{E\}\; [H]\; =\; langle\; H\; angle\; =\; -frac\; \{partial\; log(Z(eta))\}\; \{partial\; eta\}$

gives the average (expectation value) of "H". In physics, this would be called the average

energy of the system.The entropy is given by

:$egin\{align\}\; S\; =\; -sum\_\{x\_i\}\; P(x\_1,x\_2,dots)\; ln\; P(x\_1,x\_2,dots)\; \backslash \; =\; eta\; langle\; H\; angle\; +\; log\; Z(eta)end\{align\}$

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 method s.By introducing artificial auxiliary functions $J\_k$ into the partition function, it can then be used to obtain the expectation value of the random variables. Thus, for example, by writing

:$egin\{align\}\; Z(eta,J)\; =\; Z(eta,J\_1,J\_2,dots)\; \backslash \; =\; sum\_\{x\_i\}\; exp\; left(-eta\; H(x\_1,x\_2,dots)\; +sum\_n\; J\_n\; x\_n\; ight)end\{align\}$

one then has :$old\{E\}\; [x\_k]\; =\; langle\; x\_k\; angle\; =\; left.frac\{partial\}\{partial\; J\_k\}log\; Z(eta,J)\; ight|\_\{J=0\}$

as the expectation value of $x\_k$.

**Correlation functions**Multiple differentiations lead to the

correlation function s of the random variables. Thus the correlation function $C(x\_j,x\_k)$ between variables $x\_j$ and $x\_k$ is given by::$C(x\_j,x\_k)\; =\; left.frac\{partial\}\{partial\; J\_j\}frac\{partial\}\{partial\; J\_k\}log\; Z(eta,J)\; ight|\_\{J=0\}$

For the case where "H" can be written as a

quadratic form involving adifferential operator , that is, as:$H\; =\; frac\{1\}\{2\}\; sum\_n\; x\_n\; D\; x\_n$

then the correlation function $C(x\_j,x\_k)$ can be understood to be the

Green's function for the differential operator (and generally giving rise toFredholm theory ).**General properties**Partition functions often show

critical scaling , universality and are subject to therenormalization group .**ee also***

Exponential family

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