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 ${\displaystyle L}$ is a generator of evolutions for an initial state ${\displaystyle \rho _{0}}$, so that the state at any later time is given by ${\displaystyle \rho (t)=e^{Lt}[\rho _{0}]}$. Then the condition for ${\displaystyle \rho _{\infty }}$ to be a Gibbs state is

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