Partition function (mathematics): Difference between revisions

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where the sum over ''s'' is a sum over some subset of the [[power set]] ''P''(''X'') of the set <math>X=\lbrace x_1,x_2,\dots \rbrace</math>. 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 [[clique (graph theory)|cliques]] of a graph; so, for the Ising model and other [[lattice model (physics)|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 [[groupGroup action (mathematics)|action]] of a [[group (mathematics)|group symmetry]], such as [[translational invariance]]. Such symmetries can be discrete or continuous; they materialize in the [[correlation 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 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 class]]es of the symmetry. This leads to the widespread appearance of the partition function in problems with the Markov property, such as [[Hopfield network]]s.