An overview of orderings defined on the space of Markov chains having a prespecified unique stationary distribution is given. The intuition gained by studying these orderings is used to improve existing Markov chain Monte Carlo algorithms.
Antonietta Mira. "Ordering and Improving the Performance of Monte Carlo Markov Chains." Statist. Sci. 16 (4) 340 - 350, November 2001. https://doi.org/10.1214/ss/1015346319