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| In [[mathematics]], the '''Gibbs measure''', named after [[Josiah Willard Gibbs]], is a [[probability measure]] frequently seen in many problems of [[probability theory]] and [[statistical mechanics]]. It is the measure associated with the [[canonical ensemble]]. Gibbs measure implies the [[Markov property]] (a certain kind of statistical independence); and importantly, it implies the [[Hammersley–Clifford theorem]] that the energy function can be written as a multiplication of parts, thus leading to its widespread appearance in many problems outside of [[physics]], such as [[Hopfield network]]s, [[Markov network]]s, and [[Markov logic network]]s. In addition, the Gibbs measure is the unique measure that maximizes the [[entropy (general concept)|entropy]] for a given expected energy; thus, the Gibbs measure underlies [[maximum entropy method]]s and the algorithms derived therefrom.
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| The measure gives the probability of the system ''X'' being in state ''x'' (equivalently, of the [[random variable]] ''X'' having value ''x'') as
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| :<math>P(X=x) = \frac{1}{Z(\beta)} \exp \left( - \beta E(x) \right).</math>
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| Here, <math>E(x)</math> is a function from the space of states to the real numbers; in physics applications, <math>E(x)</math> is interpreted as the energy of the configuration ''x''. The parameter <math>\beta</math> is a free parameter; in physics, it is the [[inverse temperature]]. The [[normalizing constant]] <math>Z(\beta)</math> is the [[partition function (mathematics)|partition function]].
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| ==Markov property==
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| An example of the [[Markov property]] of the Gibbs measure can be seen in the [[Ising model]]. Here, the probability of a given spin <math>\sigma_k</math> being in state ''s'' is, in principle, dependent on all other spins in the model; thus one writes
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| :<math>P(\sigma_k = s|\sigma_j,\, j\ne k)</math>
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| for this probability. However, the interactions in the Ising model are nearest-neighbor interactions, and thus, one actually has
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| :<math>P(\sigma_k = s|\sigma_j,\, j\ne k) =
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| P(\sigma_k = s|\sigma_j,\, j\isin N_k)
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| </math>
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| where <math>N_k</math> is the set of nearest neighbors of site <math>k</math>. That is, the probability at site <math>k</math> depends ''only'' on the nearest neighbors. This last equation is in the form of a Markov-type statistical independence. Measures with this property are sometimes called [[Markov random field]]s. More strongly, the converse is also true: ''any'' positive probability distribution (non-zero everywhere) having the Markov property can be represented with the Gibbs measure, given an appropriate energy function;<ref>Ross Kindermann and J. Laurie Snell, [http://www.ams.org/online_bks/conm1/ Markov Random Fields and Their Applications] (1980) American Mathematical Society, ISBN 0-8218-5001-6</ref> this is the [[Hammersley–Clifford theorem]]. | |
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| ==Gibbs measure on lattices==
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| What follows is a formal definition for the special case of a random field on a group lattice. The idea of a Gibbs measure is, however, much more general than this.
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| The definition of a '''Gibbs random field''' on a [[lattice (group)|lattice]] requires some terminology:
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| * The '''lattice''': A countable set <math>\mathbb{L}</math>.
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| * The '''single-spin space''': A [[probability space]] <math>(S,\mathcal{S},\lambda)</math>.
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| * The '''[[configuration space]]''': <math>(\Omega, \mathcal{F})</math>, where <math>\Omega = S^{\mathbb{L}}</math> and <math>\mathcal{F} = \mathcal{S}^{\mathbb{L}}</math>.
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| * Given a configuration <math>\omega \in \Omega</math> and a subset <math>\Lambda \subset \mathbb{L}</math>, the restriction of <math>\omega</math> to <math>\Lambda</math> is <math>\omega_\Lambda = (\omega(t))_{t\in\Lambda}</math>. If <math>\Lambda_1\cap\Lambda_2=\emptyset</math> and <math>\Lambda_1\cup\Lambda_2=\mathbb{L}</math>, then the configuration <math>\omega_{\Lambda_1}\omega_{\Lambda_2}</math> is the configuration whose restrictions to <math>\Lambda_1</math> and <math>\Lambda_2</math> are <math>\omega_{\Lambda_1}</math> and <math>\omega_{\Lambda_2}</math>, respectively. These will be used to define [[cylinder set]]s, below.
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| * The set <math>\mathcal{L}</math> of all finite subsets of <math>\mathbb{L}</math>.
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| * For each subset <math>\Lambda\subset\mathbb{L}</math>, <math>\mathcal{F}_\Lambda</math> is the [[sigma algebra|<math>\sigma</math>-algebra]] generated by the family of functions <math>(\sigma(t))_{t\in\Lambda}</math>, where <math>\sigma(t)(\omega)=\omega(t)</math>. This sigma-algebra is just the algebra of [[cylinder set]]s on the lattice.
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| * The '''[[potential]]''': A family <math>\Phi=(\Phi_A)_{A\in\mathcal{L}}</math> of functions <math>\Phi_A:\Omega \to \mathbb{R}</math> such that
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| *# For each <math>A\in\mathcal{L}</math>, <math>\Phi_A</math> is <math>\mathcal{F}_A</math>-[[measurable]].
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| *# For all <math>\Lambda\in\mathcal{L}</math> and <math>\omega\in\Omega</math>, the series <math>H_\Lambda^\Phi(\omega) = \sum_{A\in\mathcal{L}, A\cap\Lambda\neq\emptyset} \Phi_A(\omega)</math> exists.
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| * The '''[[Hamiltonian mechanics#Mathematical formalism|Hamiltonian]]''' in <math>\Lambda\in\mathcal{L}</math> with '''boundary conditions''' <math>\bar\omega</math>, for the potential <math>\Phi</math>, is defined by
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| <center><math>H_\Lambda^\Phi(\omega | \bar\omega) = H_\Lambda^\Phi(\omega_\Lambda\bar\omega_{\Lambda^c})</math>,</center>
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| :where <math>\Lambda^c = \mathbb{L}\setminus\Lambda</math>.
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| * The '''[[partition function (mathematics)|partition function]]''' in <math>\Lambda\in\mathcal{L}</math> with '''boundary conditions''' <math>\bar\omega</math> and inverse temperature <math>\beta\in\mathbb{R}_+</math> (for the potential <math>\Phi</math> and <math>\lambda</math>) is defined by
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| <center><math>Z_\Lambda^\Phi(\bar\omega) = \int \lambda^\Lambda(\mathrm{d}\omega) \exp(-\beta H_\Lambda^\Phi(\omega | \bar\omega))</math>.</center>
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| :Here <math>\lambda^\Lambda(\mathrm{d}\omega)</math> is the product measure <math>\prod_{t\in\Lambda}\lambda(\mathrm{d}\omega(t))</math>.
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| :A potential <math>\Phi</math> is <math>\lambda</math>-admissible if <math>Z_\Lambda^\Phi(\bar\omega)</math> is finite for all <math>\Lambda\in\mathcal{L}</math>, <math>\bar\omega\in\Omega</math> and <math>\beta>0</math>.
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| A [[probability measure]] <math>\mu</math> on <math>(\Omega,\mathcal{F})</math> is a '''Gibbs measure''' for a <math>\lambda</math>-admissible potential <math>\Phi</math> if it satisfies the '''[[Dobrushin-Lanford-Ruelle]] (DLR) equations'''
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| <center><math>\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)</math>,</center>
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| : for all <math>A\in\mathcal{F}</math> and <math>\Lambda\in\mathcal{L}</math>.
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| ===An example===
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| 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 <math>J</math>) and a magnetic field (<math>h</math>), on <math>\mathbb{Z}^d</math>:
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| * The lattice is simply <math>\mathbb{L} = \mathbb{Z}^d</math>.
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| * The single-spin space is <math>S=\{-1,1\}</math>.
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| * The potential is given by
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| <center><math>\Phi_A(\omega) = \begin{cases}
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| -J\,\omega(t_1)\omega(t_2) & \mathrm{if\ } A=\{t_1,t_2\} \mathrm{\ with\ } \|t_2-t_1\|_1 = 1 \\
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| -h\,\omega(t) & \mathrm{if\ } A=\{t\}\\
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| 0 & \mathrm{otherwise}
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| \end{cases}</math></center>
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| ==See also==
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| * [[Exponential family]]
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| * [[Gibbs algorithm]]
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| * [[Gibbs sampling]]
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| * [[Interacting particle system]]
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| * [[Stochastic cellular automata]]
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| ==References==
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| <references/>
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| * Georgii, H.-O. "Gibbs measures and phase transitions", de Gruyter, Berlin, 1988, 2nd edition 2011.
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| {{Stochastic processes}}
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| [[Category:Measures (measure theory)]]
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| [[Category:Statistical mechanics]]
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I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Alaska is exactly where I've always been living. She is truly fond of caving but she doesn't have the time recently. Distributing manufacturing is exactly where my main income arrives from and it's some thing I truly enjoy.
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