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 $L \;$ is a generator of evolutions for an initial state $\rho_0 \;$, so that the state at any later time is given by $\rho(t) = e^{L t} [\rho_0] \;$. Then the condition for $\rho_{\infty} \;$ to be a Gibbs state is

$L [\rho_{\infty}] = 0$ .

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.

See also

• Gibbs algorithm
• Gibbs measure