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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}


  • Georgii, H.-O. "Gibbs measures and phase transitions", de Gruyter, Berlin, 1988.
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Gibbs_measure". A list of authors is available in Wikipedia.
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