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# Gibbs measure

In statistical mechanics, a Gibbs measure is a probability measure that relates the probabilities of the various possible states of a system to the energies associated to them. Although the precise definition requires some care (particularly in the case of infinite systems), the main characteristic of a Gibbs measure is that the probability of the system assuming a given state ω with associated energy E(ω) at inverse temperature β is proportional to

$\exp \left( - \beta E(\omega) \right).$

## Formal definition

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

• The lattice: A countable set $\mathbb{L}$.
• The single-spin space: A probability space $(S,\mathcal{S},\lambda)$.
• The configuration space: $(\Omega, \mathcal{F})$, where $\Omega = S^{\mathbb{L}}$ and $\mathcal{F} = \mathcal{S}^{\mathbb{L}}$.
• Given a configuration $\omega \in \Omega$ and a subset $\Lambda \subset \mathbb{L}$, the restriction of ω to Λ is $\omega_\Lambda = (\omega(t))_{t\in\Lambda}$. If $\Lambda_1\cap\Lambda_2=\emptyset$ and $\Lambda_1\cup\Lambda_2=\mathbb{L}$, then the configuration $\omega_{\Lambda_1}\omega_{\Lambda_2}$ is the configuration whose restrictions to Λ1 and Λ2 are $\omega_{\Lambda_1}$ and $\omega_{\Lambda_2}$, respectively.
• The set $\mathcal{L}$ of all finite subsets of $\mathbb{L}$.
• For each subset $\Lambda\subset\mathbb{L}$, $\mathcal{F}_\Lambda$ is the σ-algebra generated by the family of functions $(\sigma(t))_{t\in\Lambda}$, where σ(t)(ω) = ω(t).
• The potential: A family $\Phi=(\Phi_A)_{A\in\mathcal{L}}$ of functions $\Phi_A:\Omega \to \mathbb{R}$ such that
1. For each $A\in\mathcal{L}$, ΦA is $\mathcal{F}_A$-measurable.
2. For all $\Lambda\in\mathcal{L}$ and $\omega\in\Omega$, the series $H_\Lambda^\Phi(\omega) = \sum_{A\in\mathcal{L}, A\cap\Lambda\neq\emptyset} \Phi_A(\omega)$ exists.
• The Hamiltonian in $\Lambda\in\mathcal{L}$ with boundary conditions $\bar\omega$, for the potential Φ, is defined by
$H_\Lambda^\Phi(\omega | \bar\omega) = H_\Lambda^\Phi(\omega_\Lambda\bar\omega_{\Lambda^c})$,
where $\Lambda^c = \mathbb{L}\setminus\Lambda$.
• The partition function in $\Lambda\in\mathcal{L}$ with boundary conditions $\bar\omega$ and inverse temperature $\beta\in\mathbb{R}_+$ (for the potential Φ and λ) is defined by
$Z_\Lambda^\Phi(\bar\omega) = \int \lambda^\Lambda(\mathrm{d}\omega) \exp(-\beta H_\Lambda^\Phi(\omega | \bar\omega))$.
A potential Φ is λ-admissible if $Z_\Lambda^\Phi(\bar\omega)$ is finite for all $\Lambda\in\mathcal{L}$, $\bar\omega\in\Omega$ and β > 0.

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

$\int \mu(\mathrm{d}\bar\omega)Z_\Lambda^\Phi(\bar\omega)^{-1} \int\lambda^\Lambda(\mathrm{d}\omega) \exp(-\beta H_\Lambda^\Phi(\omega | \bar\omega)) 1_A(\omega_\Lambda\bar\omega_{\Lambda^c}) = \mu(A)$,
for all $A\in\mathcal{F}$ and $\Lambda\in\mathcal{L}$.

### An example

To help understand the above definitions, here are the corresponding quantities in the important example of the Ising model with nearest-neighbour interactions (coupling constant J) and a magnetic field (h), on $\mathbb{Z}^d$:

• The lattice is simply $\mathbb{L} = \mathbb{Z}^d$.
• The single-spin space is S = { − 1,1}.
• The potential is given by
$\Phi_A(\omega) = \begin{cases} -J\,\omega(t_1)\omega(t_2) & \mathrm{if\ } A=\{t_1,t_2\} \mathrm{\ with\ } \|t_2-t_1\|_1 = 1 \\ -h\,\omega(t) & \mathrm{if\ } A=\{t\}\\ 0 & \mathrm{otherwise} \end{cases}$

## References

• Georgii, H.-O. "Gibbs measures and phase transitions", de Gruyter, Berlin, 1988.