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As the title says, what would be non-ergodic processes that occur in statistical physics? Many textbooks do not really cover ergodicity really well so I ask this question. I can't suddenly remember any non-ergodic process in physics..

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2 Answers 2

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A process is ergodic if you get the same statistical momenta

  • considering a single realization for a sufficiently long time,

and by

  • considering a sufficiently big number of realizations at a precise moment.

In other word you can extract all the information on a process by looking at a single realization for some time. Having said that is clear that any non-stationary process cannot be ergodic (different realizations will behave differently as time goes), but we can have non ergodic stationary processes. Let's see some examples:

  1. Every second you flip a coin and you sample the result: this is a stationary process and also ergodic. If you had flipped $N$ coins at $t=0$ you would have got the same result.
  2. You flip a coin once at $t=0$ and then you sample the result every second: this is a stationary process, but not ergodic. If you had flipped $N$ coins and sampled at $t=0$ you would have got a different mean with respect to the one that you get in time.
  3. You keep flipping the coin and sampling every second until you get head, then stop flipping, but go on sampling. This process is not stationary and not ergodic as well, although both the properties are approached going to big times.
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  • $\begingroup$ Could you clarify $3$? The trajectory eventually becomes heads while sampling at $0$ varies. $\endgroup$
    – A.S.
    Dec 21, 2015 at 4:57
  • $\begingroup$ @A.S. I do not understand what is not clear... $\endgroup$
    – DarioP
    Dec 23, 2015 at 22:41
  • $\begingroup$ In what sense does a process $(3)$ approach stationarity and ergodicity going to "big times"? It becomes constant (head) eventually - so do you stop comparing to the first flip and compare to some other "distant" flip? $\endgroup$
    – A.S.
    Dec 23, 2015 at 22:57
  • $\begingroup$ @A.S. Then it's quite a matter of definition. In the definition that I used the sampling of multiple realizations doesn't need to be taken at $t=0$ but at a generic precise moment. You could take it at $t\rightarrow+\infty$, then you have a constant process. In a stronger definition you may require that for any time, then this wouldn't work. $\endgroup$
    – DarioP
    Dec 23, 2015 at 23:10
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Motion of integrable systems, which are confined to their KAM tori. They don't sample all of phase space, and in fact if they have rational tunes they don't even sample "all of the KAM torus".

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