In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, a Gibbs state is an equilibrium probability distribution which remains invariant under future evolution of the system. For example, a stationary or steady-state distribution of a
Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
, such as that achieved by running a
Markov chain Monte Carlo
In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain ...
iteration for a sufficiently long time, is a Gibbs state.
Precisely, suppose
is a generator of evolutions for an initial state
, so that the state at any later time is given by
. Then the condition for
to be a Gibbs state is
:
.
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
there may be several physically distinct Gibbs states in which a system may be trapped, particularly at lower temperatures.
They are named after
Josiah Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
, for his work in determining equilibrium properties of
statistical ensemble
In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...
s. Gibbs himself referred to this type of statistical ensemble as being in "statistical equilibrium".
See also
*
Gibbs algorithm
200px, Josiah Willard Gibbs
In statistical mechanics, the Gibbs algorithm, introduced by J. Willard Gibbs in 1902, is a criterion for choosing a probability distribution for the statistical ensemble of microstates of a thermodynamic system by ...
*
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.
Th ...
*
KMS state
In the statistical mechanics of quantum mechanical systems and quantum field theory, the properties of a system in thermal equilibrium can be described by a mathematical object called a Kubo–Martin– Schwinger state or, more commonly, a KMS s ...
References
Statistical mechanics
Stochastic processes
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