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
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 configuration x. The parameter β is a free parameter; in physics, it is the
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
The
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 as symmetry breaking and phase coexistence.
Statistical physics
The set of Gibbs measures on a system is always convex,
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 typically not invariant under the Hamiltonian's symmetry. For example, in the infinite ferromagnetic Ising model below the critical temperature, there are two pure states, the "mostly-up" and "mostly-down" states, which are interchanged under the model's symmetry.
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 σk 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
- .
However, in an Ising model with only finite-range interactions (for example, nearest-neighbor interactions), we actually have
- ,
where Nk 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.[2] 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 .
- The single-spin space: A probability space .
- The configuration space: , where and .
- Given a configuration ω ∈ Ω and a subset , the restriction of ω to Λ is . If and , then the configuration is the configuration whose restrictions to Λ1 and Λ2 are and , respectively.
- The set of all finite subsets of .
- For each subset , is the σ-algebragenerated by the family of functions , where . The union of these σ-algebras as varies over is the algebra of cylinder sets on the lattice.
- The potential: A family of functions ΦA : Ω → R such that
- For each is -measurable, meaning it depends only on the restriction (and does so measurably).
- For all and ω ∈ Ω, the following series exists:[when defined as?]
- For each is -
We interpret ΦA as the contribution to the total energy (the Hamiltonian) associated to the interaction among all the points of finite set A. Then as the contribution to the total energy of all the finite sets A that meet . Note that the total energy is typically infinite, but when we "localize" to each it may be finite, we hope.
- The Hamiltonian in with boundary conditions , for the potential Φ, is defined by
- where denotes the configuration taking the values of in , and those of in .
- The partition function in with boundary conditions and inverse temperature β > 0 (for the potential Φ and λ) is defined by
- where
- is the product measure
- A potential Φ is λ-admissible if is finite for all and β > 0.
- A probability measure μ on is a Gibbs measure for a λ-admissible potential Φ if it satisfies the Dobrushin–Lanford–Ruelle (DLR) equation
- for all and .
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 Zd:
- The lattice is simply .
- The single-spin space is S = {−1, 1}.
- The potential is given by
See also
- Boltzmann distribution
- Exponential family
- Gibbs algorithm
- Gibbs sampling
- Interacting particle system
- Potential game
- Softmax
- Stochastic cellular automata
References
- ^ "Gibbs measures" (PDF).
- ISBN 0-8218-5001-6
Further reading
- 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.