Gibbs state
In probability theory and statistical mechanics, 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, such as that achieved by running a Markov chain Monte Carlo 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 ![\rho(t) = e^{L t} [\rho_0] \;](/2012-wikipedia_en_all_nopic_01_2012/I/d08bb637e5825bae0c061c1bed1838d7.png)
. Then the condition for 
to be a Gibbs state is
![L [\rho_{\infty}] = 0](/2012-wikipedia_en_all_nopic_01_2012/I/6381807d58a3130faa7ef0769d082170.png)
.
In physics there may be several physically distinct Gibbs states in which a system may be trapped, particularly at lower temperatures.
They are named after J. Willard Gibbs, for his work in determining equilibrium properties of statistical ensembles.