A non-binary measure of ergodicity

Question

All definitions of ergodicity that I can find state it that a system is either ergodic or not (aka it's binary).

$<X_t> = <X_e>$ Basically it's ergodic where the time average equals the ensemble average and not if it doesn't.

What I'm interested in is: is there any notion or definition of "how much a system is ergodic"? You might say a "measure of ergodic-ness"?

For instance something like $\dfrac{<X_t>-<X_e>}{<X_e>}$ but far less naive.

Answer 1

Ergodicity means that the system over time will eventually reach all possible states. If that is true, then for any observable, the time and ensemble average will be the same. But if the system is not ergodic you can probably still find some observables for which that is true.

So it seems to me that you have two options:

• For an ergodicity-measure for the system itself: Compute the fraction of the total state-space that a single trajectory will visit over time.
• For a measure depending on a conrete observable: Consider $X_e$ and $X_t$ as two random variables. The first is random over the whole state space, and the second is a random time in a single given trajectory. Comparing two random variables can be done in many different ways, see wikipedia for example.

As a side note: Even if a process is ergodic in theory, it may take arbitrarily long for it to visit all parts of the state space. This is a problem if you run a simulation for a finite amount of time, so you are not sure if you covered the state-space sufficiently to assume that the finite-time-average is close to the true ensemble-average. Here the concept of "autocorrelation" is useful. In particular the "integrated autocorrelation time" gives you a measure for how fast your system is moving through state space (and an infinite autocorrelation time means your system is not ergodic).