One way to monitor whether a chain has converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. This is the motivation for the Gelman and Rubin potential scale reduction statistic Rhat. The Rhat statistic measures  the ratio of the average variance of samples within each chain to the variance of the pooled samples across chains; if all chains are at equilibrium, these will be the same and Rhat will be one. If the chains have not converged to a common distribution, the Rhat statistic will be greater than one. See the Glossary page for more details. 