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) asThe set of Gibbs measures on a system is always convex, 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 of convex combinations of a much smaller number of special Gibbs measures known as 'pure states' (not to be confused with the related but distinct notion of pure states in quantum mechanics). In physical applications, the Hamiltonian (the energy function) usually has some sense of locality, and the pure states have the cluster decomposition property that 'far-separated subsystems' are independent. In practice, physically realistic systems are found in one of these pure states.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 asWhat 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.