Partition function (mathematics)

From Wikipedia, the free encyclopedia

The partition function, as used in probability theory, information science and dynamical systems, is an abstraction of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory.

[edit] Definition

Given a set of random variables $x i$ and some sort of of potential function or Hamiltonian $H(x_1,x_2,\dots)$ , the partition function is defined as

$Z(\beta) = \sum_{x_i} \exp \left(-\beta H(x_1,x_2,\dots) \right)$

The sume over the $x i$ is understood to be a sum over all possible values that the $x i$ may take.

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

$H(x_1,x_2,\dots) = \sum_s \varphi_s$

where the sum over s is a sum over some subset of the power set P(X) of the set $X=\lbrace x_1,x_2,\dots \rbrace$ . 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.

[edit] Exposition

The value of the expression

$\exp \left(-\beta H(x_1,x_2,\dots) \right)$

can be interpreted as a likelihood that a specific configuration of values $(x_1,x_2,\dots)$ occurs in the system. The sum over the $x i$ is then a sum over all possible configurations. Thus, given a specific configuration $(x_1,x_2,\dots)$ ,

$P(x_1,x_2,\dots) = \frac{1}{Z(\beta)} \exp \left(-\beta H(x_1,x_2,\dots) \right)$

is the probability of the configuration $(x_1,x_2,\dots)$ occurring in the system, which is now properly normalized so that $0\le P(x_1,x_2,\dots)\le 1$ , 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.

The set of variables $x i$ need not be countable, in which case, the sums are to be replaced by integrals. The value $β$ need not be real; both these cases occur in the definition of the partition function in quantum field theory.

The value of $β$ is typically taken to be an adjustable parameter of the theory, in which case derivatives of $Z (β)$ with respect to $β$ can have significant, meaningful interpretations. For example,

$\langle H \rangle = \frac {\partial Z (\beta)} {\partial \beta}$