Gibbs measure

The set of Gibbs measures on a system is always convex,[1] so there is either a unique Gibbs measure (in which case the system is said to be "ergodic"), or there are infinitely many (and the system is called "nonergodic"). In the nonergodic case, the Gibbs measures can be expressed as the set of convex combinations of a much smaller number of special Gibbs measures known as "pure states" (not to be confused with the related but distinct notion of pure states in quantum mechanics). In physical applications, the Hamiltonian (the energy function) usually has some sense of locality, and the pure states have the cluster decomposition property that "far-separated subsystems" are independent. In practice, physically realistic systems are found in one of these pure states.

If the Hamiltonian possesses a symmetry, then a unique (i.e. ergodic) Gibbs measure will necessarily be invariant under the symmetry. But in the case of multiple (i.e. nonergodic) Gibbs measures, the pure states are typically not invariant under the Hamiltonian's symmetry. For example, in the infinite ferromagnetic Ising model below the critical temperature, there are two pure states, the "mostly-up" and "mostly-down" states, which are interchanged under the model's ${\displaystyle \mathbb {Z} _{2}}$ symmetry.

An example of the Markov property can be seen in the Gibbs measure of the Ising model. The probability for a given spin σ_k to be in state s could, in principle, depend on the states of all other spins in the system. Thus, we may write the probability as

${\displaystyle P(\sigma _{k}=s\mid \sigma _{j},\,j\neq k)}$.

However, in an Ising model with only finite-range interactions (for example, nearest-neighbor interactions), we actually have

${\displaystyle P(\sigma _{k}=s\mid \sigma _{j},\,j\neq k)=P(\sigma _{k}=s\mid \sigma _{j},\,j\in N_{k})}$,

where N_k is a neighborhood of the site k. That is, the probability at site k depends only on the spins in a finite neighborhood. This last equation is in the form of a local Markov property. Measures with this property are sometimes called Markov random fields. More strongly, the converse is also true: any positive probability distribution (nonzero density everywhere) having the Markov property can be represented as a Gibbs measure for an appropriate energy function.[2] This is the Hammersley–Clifford theorem.

What follows is a formal definition for the special case of a random field on a lattice. The idea of a Gibbs measure is, however, much more general than this.

The definition of a Gibbs random field on a lattice requires some terminology:

• The lattice: A countable set ${\displaystyle \mathbb {L} }$.
• The single-spin space: A probability space ${\displaystyle (S,{\mathcal {S}},\lambda )}$.
• The configuration space: ${\displaystyle (\Omega ,{\mathcal {F}})}$, where ${\displaystyle \Omega =S^{\mathbb {L} }}$ and ${\displaystyle {\mathcal {F}}={\mathcal {S}}^{\mathbb {L} }}$.
• Given a configuration ω ∈ Ω and a subset ${\displaystyle \Lambda \subset \mathbb {L} }$, the restriction of ω to Λ is ${\displaystyle \omega _{\Lambda }=(\omega (t))_{t\in \Lambda }}$. If ${\displaystyle \Lambda _{1}\cap \Lambda _{2}=\emptyset }$ and ${\displaystyle \Lambda _{1}\cup \Lambda _{2}=\mathbb {L} }$, then the configuration ${\displaystyle \omega _{\Lambda _{1}}\omega _{\Lambda _{2}}}$ is the configuration whose restrictions to Λ₁ and Λ₂ are ${\displaystyle \omega _{\Lambda _{1}}}$ and ${\displaystyle \omega _{\Lambda _{2}}}$, respectively.
• The set ${\displaystyle {\mathcal {L}}}$ of all finite subsets of ${\displaystyle \mathbb {L} }$.
• For each subset ${\displaystyle \Lambda \subset \mathbb {L} }$, ${\displaystyle {\mathcal {F}}_{\Lambda }}$ is the σ-algebra generated by the family of functions ${\displaystyle (\sigma (t))_{t\in \Lambda }}$, where ${\displaystyle \sigma (t)(\omega )=\omega (t)}$. The union of these σ-algebras as ${\displaystyle \Lambda }$ varies over ${\displaystyle {\mathcal {L}}}$ is the algebra of cylinder sets on the lattice.
• The potential: A family ${\displaystyle \Phi =(\Phi _{A})_{A\in {\mathcal {L}}}}$ of functions Φ_A : Ω → R such that
  1. For each ${\displaystyle A\in {\mathcal {L}},\Phi _{A}}$ is ${\displaystyle {\mathcal {F}}_{A}}$-measurable, meaning it depends only on the restriction ${\displaystyle \omega _{A}}$ (and does so measurably).
  2. For all ${\displaystyle \Lambda \in {\mathcal {L}}}$ and ω ∈ Ω, the following series exists:

${\displaystyle H_{\Lambda }^{\Phi }(\omega )=\sum _{A\in {\mathcal {L}},A\cap \Lambda \neq \emptyset }\Phi _{A}(\omega ).}$

We interpret Φ_A as the contribution to the total energy (the Hamiltonian) associated to the interaction among all the points of finite set A. Then ${\displaystyle H_{\Lambda }^{\Phi }(\omega )}$ as the contribution to the total energy of all the finite sets A that meet ${\displaystyle \Lambda }$. Note that the total energy is typically infinite, but when we "localize" to each ${\displaystyle \Lambda }$ it may be finite, we hope.

• The Hamiltonian in ${\displaystyle \Lambda \in {\mathcal {L}}}$ with boundary conditions ${\displaystyle {\bar {\omega }}}$, for the potential Φ, is defined by

${\displaystyle H_{\Lambda }^{\Phi }(\omega \mid {\bar {\omega }})=H_{\Lambda }^{\Phi }\left(\omega _{\Lambda }{\bar {\omega }}_{\Lambda ^{c}}\right)}$

where ${\displaystyle \Lambda ^{c}=\mathbb {L} \setminus \Lambda }$.

• The partition function in ${\displaystyle \Lambda \in {\mathcal {L}}}$ with boundary conditions ${\displaystyle {\bar {\omega }}}$ and inverse temperature β > 0 (for the potential Φ and λ) is defined by

${\displaystyle Z_{\Lambda }^{\Phi }({\bar {\omega }})=\int \lambda ^{\Lambda }(\mathrm {d} \omega )\exp(-\beta H_{\Lambda }^{\Phi }(\omega \mid {\bar {\omega }})),}$

where

${\displaystyle \lambda ^{\Lambda }(\mathrm {d} \omega )=\prod _{t\in \Lambda }\lambda (\mathrm {d} \omega (t)),}$

is the product measure

A potential Φ is λ-admissible if ${\displaystyle Z_{\Lambda }^{\Phi }({\bar {\omega }})}$ is finite for all ${\displaystyle \Lambda \in {\mathcal {L}},{\bar {\omega }}\in \Omega }$ and β > 0.

A probability measure μ on ${\displaystyle (\Omega ,{\mathcal {F}})}$ is a Gibbs measure for a λ-admissible potential Φ if it satisfies the Dobrushin–Lanford–Ruelle (DLR) equation

${\displaystyle \int \mu (\mathrm {d} {\bar {\omega }})Z_{\Lambda }^{\Phi }({\bar {\omega }})^{-1}\int \lambda ^{\Lambda }(\mathrm {d} \omega )\exp(-\beta H_{\Lambda }^{\Phi }(\omega \mid {\bar {\omega }}))1_{A}(\omega _{\Lambda }{\bar {\omega }}_{\Lambda ^{c}})=\mu (A),}$

for all ${\displaystyle A\in {\mathcal {F}}}$ and ${\displaystyle \Lambda \in {\mathcal {L}}}$.

• Georgii, H.-O. (2011) [1988]. Gibbs Measures and Phase Transitions (2nd ed.). Berlin: de Gruyter. ISBN 978-3-11-025029-9.
• Friedli, S.; Velenik, Y. (2017). Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction. Cambridge: Cambridge University Press. ISBN 9781107184824.