²¹There exist algorithms which are non-ergodic but the non reachable states are of “measure zero” in the space of states. These algorithms are formally non ergodic, but they are ergodic from a practical point of view. On the contrary, there exist algorithms that are formally ergodic but there are large regions of phase space where the probability of getting there is very small. This puts “ergodic barriers” in the sampling which will lead to wrong results. A common example is sampling a system in the neighborhood of a first order phase transition where, for large systems, it is very hard to sample states in both phases.