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 the measure associated with the Boltzmann distribution, and generalizes the notion of the canonical ensemble. Importantly, when the energy function can be written as a sum of parts, the Gibbs measure has the Markov property (a certain kind of statistical independence), thus leading to its widespread appearance in many problems outside of physics, such as Hopfield networks, Markov networks, and Markov logic networks. In addition, the Gibbs measure is the unique measure that maximizes the entropy for a given expected energy; thus, the Gibbs measure underlies maximum entropy methods and the algorithms derived therefrom.

The measure gives the probability of the system X being in state x (equivalently, of the random variable X having value x) as

Here, is a function from the space of states to the real numbers; in physics applications, is interpreted as the energy of the configuration x. The parameter is a free parameter; in physics, it is the inverse temperature. The normalizing constant is the partition function.