Gibbs measure

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 a generalization of the canonical ensemble to infinite systems. The canonical ensemble gives the probability of the system X being in state x (equivalently, of the random variable X having value x) as

P(X=x) = \frac{1}{Z(\beta)} \exp ( - \beta E(x)).

Here, $E (x)$ is a function from the space of states to the real numbers; in physics applications, $E (x)$ is interpreted as the energy of the configuration x. The parameter $β$ is a free parameter; in physics, it is the inverse temperature. The normalizing constant $Z (β)$ is the partition function. However, in infinite systems, the total energy is no longer a finite number and cannot be used in the traditional construction of the probability distribution of a canonical ensemble. Traditional approaches in statistical physics studied the limit of intensive properties as the size of a finite system approaches infinity (the thermodynamic limit). When the energy function can be written as a sum of terms that each involve only variables from a finite subsystem, the notion of a Gibbs measure provides an alternative approach. Gibbs measures were proposed by probability theorists such as Dobrushin, Lanford, and Ruelle and provided a framework to directly study infinite systems, instead of taking the limit of finite systems.

A measure is a Gibbs measure if the conditional probabilities it induces on each finite subsystem satisfy a consistency condition: if all degrees of freedom outside the finite subsystem are frozen, the canonical ensemble for the subsystem subject to these boundary conditions matches the probabilities in the Gibbs measure conditional on the frozen degrees of freedom.

The Hammersley–Clifford theorem implies that any probability measure that satisfies a Markov property is a Gibbs measure for an appropriate choice of (locally defined) energy function. Therefore, the Gibbs measure applies to widespread problems outside of physics, such as Hopfield networks, Markov networks, and Markov logic networks. A Gibbs measure in a system with local (finite-range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density.

The Gibbs measure of an infinite system is not necessarily unique, in contrast to the canonical ensemble of a finite system, which is unique. The existence of more than one Gibbs measures is associated with statistical phenomena such as symmetry breaking and phase coexistence.

Markov property

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

P(\sigma_k = s\mid\sigma_j,\, j\ne k)

.

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

P(\sigma_k = s\mid\sigma_j,\, j\ne k) = P(\sigma_k = s\mid\sigma_j,\, j\isin 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.^[1] This is the Hammersley–Clifford theorem.

Formal definition on lattices

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 $\mathbb{L}$ .
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 $ω \in Ω$ 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 $Λ 1$ and $Λ 2$ are $\omega_{\Lambda_1}$ and $\omega_{\Lambda_2}$ , respectively. These will be used to define cylinder sets, below.

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 $\sigma(t)(\omega)=\omega(t)$ . This $σ$ -algebra is just the $σ$ -algebra of cylinder sets on the lattice.
The potential: A family of functions Φ_A : Ω → R such that
1. For each $A\in\mathcal{L}, \Phi_A$ is $\mathcal{F}_A$ -measurable.
2. For all $\Lambda\in\mathcal{L}$ and $ω \in Ω$ , the following series exists:

H_\Lambda^\Phi(\omega) = \sum_{A\in\mathcal{L}, A\cap\Lambda\neq\emptyset} \Phi_A(\omega).

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

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

where

\Lambda^c = \mathbb{L}\setminus\Lambda

.

The partition function in $\Lambda\in\mathcal{L}$ with boundary conditions $\bar\omega$ and inverse temperature $β > 0$ (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 \mid \bar\omega)),

where

\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

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) equation

\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

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-neighbor interactions (coupling constant $J$ ) and a magnetic field ( $h$ ), on $Z d$ :

The lattice is simply $\mathbb{L} = \mathbf{Z}^d$ .
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) & \text{if } A=\{t_1,t_2\} \text{ with } \|t_2-t_1\|_1 = 1 \\ -h\,\omega(t) & \text{if } A=\{t\}\\ 0 & \text{otherwise} \end{cases}

References

↑ Ross Kindermann and J. Laurie Snell, Markov Random Fields and Their Applications (1980) American Mathematical Society, ISBN 0-8218-5001-6

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Gibbs measure

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Markov property

Formal definition on lattices

An example

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