# What is the definition of ergodicity in Monte Carlo?

I've been writing a manuscript about the breakdown of ergodicity in single spin flip Metropolis algorithm Monte Carlo arXiv:2001.09268.

The definition I have been using is: a Markov process is ergodic if any state $$x'$$ can be reached from any state $$x$$ in a finite number of steps.

Therefore, if the number of steps required diverges rapidly with system size, the simulation has lost ergodicity. Is that a correct interpretation? Are there competing definitions?

Ideally I would like to get a citeable source that that I can use to defend my definition.

• Related discussion: physics.stackexchange.com/questions/59761/… – Tyberius May 1 at 3:39
• @Tyberius, that discussion is useful, but I am hoping to find something in the literature that draws the distinction between ergodicity as a physics concept and ergodicity as it is refers to Monte Carlo. – taciteloquence May 27 at 8:20

Your definition is alright, and a citable reference is Landau and Binder's A Guide to Monte Carlo Simulations in Statistical Physics which says (Sec. 2.1.3):

The principle of ergodicity states that all possible configurations of the system should be attainable.

The usual definition of ergodicity, both in mathematics and physics, is that the ensemble and time averages should be the same in an ergodic system -- and in Monte Carlo simulations it's not different, except that one is often concerned with ergodicity which can be observed in practice. This is made very clear in another book by Binder, "Monte Carlo Simulation in Statistical Physics: An Introduction", which explains:

In practice one may find an apparent "breaking of ergodicity" even for systems which are ergodic, if the "time" over which the averaging is extended is not long enough, i.e., less than the so-called "ergodic time" $$\tau_e$$.

Which then continues:

There is no general rule about whether this happens or not, it really depends on the details of the algorithm.

It's also important to remember that in practice we perform simultaneously an ensemble average over the time average: where the ensemble average isn't done directly over the phase space, but comes from considering different realizations of the system (say, in the disordered case), different runs (with distinct random seeds), or different initial conditions. This is also detailed in the references above.

Another possibly interesting reference is the paper "Approach to ergodicity in Monte Carlo simulations" (e-print).