Inphysics andmathematics, theGibbs measure, named afterJosiah Willard Gibbs, is aprobability measure frequently seen in many problems ofprobability theory,statistical mechanics, anddynamical systems.^[1] It is a generalization of thecanonical ensemble to infinite systems. The canonical ensemble gives the probability of the systemX being in statex (equivalently, of therandom variableX having valuex) as

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

Here,E is a function from the space of states to the real numbers; in physics applications,E(x) is interpreted as the energy of the configurationx. The parameterβ is a free parameter; in physics, it is theinverse temperature. Thenormalizing constantZ(β) is thepartition 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 ofintensive properties as the size of a finite system approaches infinity (thethermodynamic 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 asDobrushin,Lanford, andRuelle 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 theseboundary conditions matches the probabilities in the Gibbs measureconditional on the frozen degrees of freedom.

TheHammersley–Clifford theorem implies that any probability measure that satisfies aMarkov property is a Gibbs measure for an appropriate choice of (locally defined) energy function. Therefore, the Gibbs measure applies to widespread problems outside ofphysics, such asHopfield networks,Markov networks,Markov logic networks, andboundedly rational potential games in game theory and economics. A Gibbs measure in a system with local (finite-range) interactions maximizes theentropy density for a given expectedenergy density; or, equivalently, it minimizes thefree 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 measure is associated with statistical phenomena such assymmetry breaking andphase coexistence.

The set of Gibbs measures on a system is always convex,^[2][3] 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 ofconvex combinations of a much smaller number of special Gibbs measures known as "pure states" (not to be confused with the related but distinct notion ofpure states in quantum mechanics). In physical applications, theHamiltonian (the energy function) usually has some sense oflocality, and the pure states have thecluster 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 typicallynot invariant under the Hamiltonian's symmetry. For example, in the infinite ferromagneticIsing 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 theMarkov property can be seen in the Gibbs measure of theIsing model. The probability for a given spinσ_k to be in states could, in principle, depend on the states of all other spins in the system. Thus, we may write the probability as

${\displaystyle P({\sigma }_k=s\mid {\sigma }_j,j\ne k)}$.

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

${\displaystyle P({\sigma }_k=s\mid {\sigma }_j,j\ne k)=P({\sigma }_k=s\mid {\sigma }_j,j\in N_k)}$,

whereN_k is a neighborhood of the sitek. That is, the probability at sitek dependsonly on the spins in a finite neighborhood. This last equation is in the form of a localMarkov property. Measures with this property are sometimes calledMarkov 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.^[4] This is theHammersley–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 aGibbs random field on alattice requires some terminology:

• Thelattice: A countable set${\displaystyle \mathbb{L}}$.
• Thesingle-spin space: Aprobability space${\displaystyle (S,\mathcal{S},\lambda )}$.
• Theconfiguration space:${\displaystyle (\mathrm{\Omega },\mathcal{F})}$, where${\displaystyle \mathrm{\Omega }=S^{\mathbb{L}}}$ and${\displaystyle \mathcal{F}={\mathcal{S}}^{\mathbb{L}}}$.
• Given a configurationω ∈ Ω and a subset${\displaystyle \mathrm{\Lambda }\subset \mathbb{L}}$, the restriction ofω toΛ is${\displaystyle {\omega }_{\mathrm{\Lambda }}=(\omega (t){)}_{t\in \mathrm{\Lambda }}}$. If${\displaystyle {\mathrm{\Lambda }}_1\cap {\mathrm{\Lambda }}_2=\mathrm{\varnothing }}$ and${\displaystyle {\mathrm{\Lambda }}_1\cup {\mathrm{\Lambda }}_2=\mathbb{L}}$, then the configuration${\displaystyle {\omega }_{{\mathrm{\Lambda }}_1}{\omega }_{{\mathrm{\Lambda }}_2}}$ is the configuration whose restrictions toΛ₁ andΛ₂ are${\displaystyle {\omega }_{{\mathrm{\Lambda }}_1}}$ and${\displaystyle {\omega }_{{\mathrm{\Lambda }}_2}}$, respectively.
• The set${\displaystyle \mathcal{L}}$ of all finite subsets of${\displaystyle \mathbb{L}}$.
• For each subset${\displaystyle \mathrm{\Lambda }\subset \mathbb{L}}$,${\displaystyle {\mathcal{F}}_{\mathrm{\Lambda }}}$ is theσ-algebra generated by the family of functions${\displaystyle (\sigma (t){)}_{t\in \mathrm{\Lambda }}}$, where${\displaystyle \sigma (t)(\omega )=\omega (t)}$. The union of theseσ-algebras as${\displaystyle \mathrm{\Lambda }}$ varies over${\displaystyle \mathcal{L}}$ is the algebra ofcylinder sets on the lattice.
• Thepotential: A family${\displaystyle \mathrm{\Phi }=({\mathrm{\Phi }}_A{)}_{A\in \mathcal{L}}}$ of functionsΦ_A : Ω →R such that
  1. For each${\displaystyle A\in \mathcal{L},{\mathrm{\Phi }}_A}$ is${\displaystyle {\mathcal{F}}_A}$-measurable, meaning it depends only on the restriction${\displaystyle {\omega }_A}$ (and does so measurably).
  2. For all${\displaystyle \mathrm{\Lambda }\in \mathcal{L}}$ andω ∈ Ω, the following series exists:^{[when defined as?]}

${\displaystyle H_{\mathrm{\Lambda }}^{\mathrm{\Phi }}(\omega )=\sum _{A\in \mathcal{L},A\cap \mathrm{\Lambda }\ne \mathrm{\varnothing }}{\mathrm{\Phi }}_A(\omega ).}$

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

• TheHamiltonian in${\displaystyle \mathrm{\Lambda }\in \mathcal{L}}$ withboundary conditions${\displaystyle \overline{\omega }}$, for the potentialΦ, is defined by

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

where${\displaystyle {\omega }_{\mathrm{\Lambda }}{\overline{\omega }}_{{\mathrm{\Lambda }}^c}}$ denotes the configuration taking the values of${\displaystyle \omega }$ in${\displaystyle \mathrm{\Lambda }}$, and those of${\displaystyle \overline{\omega }}$ in${\displaystyle {\mathrm{\Lambda }}^c:=\mathbb{L}\setminus \mathrm{\Lambda }}$.

• Thepartition function in${\displaystyle \mathrm{\Lambda }\in \mathcal{L}}$ withboundary conditions${\displaystyle \overline{\omega }}$ and inverse temperatureβ > 0 (for the potentialΦ andλ) is defined by

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

where

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

is the product measure

A potentialΦ isλ-admissible if${\displaystyle Z_{\mathrm{\Lambda }}^{\mathrm{\Phi }}(\overline{\omega })}$ is finite for all${\displaystyle \mathrm{\Lambda }\in \mathcal{L},\overline{\omega }\in \mathrm{\Omega }}$ andβ > 0.

Aprobability measureμ on${\displaystyle (\mathrm{\Omega },\mathcal{F})}$ is aGibbs measure for aλ-admissible potentialΦ if it satisfies theDobrushin–Lanford–Ruelle (DLR) equation

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

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

To help understand the above definitions, here are the corresponding quantities in the important example of theIsing model with nearest-neighbor interactions (coupling constantJ) and a magnetic field (h), onZ^d:

• The lattice is simply${\displaystyle \mathbb{L}={\mathbf{Z}}^d}$.
• The single-spin space isS = {−1, 1}.
• The potential is given by

• 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.
• Le Ny, A (2008).Introduction to (generalized) Gibbs measures, Ensaios Matemáticos, volume 15.
• Makhmudov, M (2025).Gibbs states in Statistical Mechanics and Dynamical Systems. PhD thesis.

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