In
probability theory
Probability theory or probability calculus 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 expre ...
and
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, a Gibbs state is an equilibrium
probability distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
which remains invariant under future evolution of the system. For example, a stationary or steady-state distribution of a
Markov chain
In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...
, such as that achieved by running a
Markov chain Monte Carlo
In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that ...
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 scientific study of matter, its Elementary particle, 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 whi ...
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 mechanical engineer and scientist who made fundamental theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynami ...
, 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
FILE:Josiah Willard Gibbs -from MMS-.jpg, 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 mi ...
*
Gibbs measure
In physics and 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 ...
*
KMS state
In the statistical mechanics of quantum mechanics, 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 (KMS) state: a ...
References
Statistical mechanics
Stochastic processes
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