A random meeting with a formal ergodic theorist

Most of the people, say more than 99.999999% of the population, has no idea of what Ergodic theory is. Therefore, if you meet someone randomly, you may be sure that he/she does not have idea at all of what it does mean. Among many other examples you can tell in order to introduce the concept of ergodicity, today,  Jana Rodríguez Hertz told us one that was quite nice and that I would not like to keep to myself.

The example Jana told us was the following:

Suppose you want to know the number of fish in a lake, say n. It is extremely difficult to count them, however you can estimate n. In order to do this, you can take p fish and mark them, then allow them to return to the lake. After some time, you take m fish and you count how many are marked. If  you assume that after enough time the proportion of marked fish is the same at “every scale”, i.e., if you take any amount k of fish you will always obtain a number r(k) of marked ones so that \frac{r(k)}{k}=\frac{p}{n}, then you can estimate n by \frac{p m}{r(m)}, where r(m) is the number of marked fish you obtained when you took in total m fish. Mathematically the validity of this statement is justified if you have the mixing property. This is a strong condition of the system.

Another strategy for estimating n is to capture m fish each 10 days, count how many are marked, and then allow them to return to the lake. Suppose the i-th time you captured fish you obtained r_i marked ones. If you do this many times, say N, then you can estimate n by \frac{Np}{\sum_{i=1}^N\frac{r_i}{m}}= \frac{Npm}{\sum_{i=1}^N r_i}.  Mathematically the validity of this statement is justified by the ergodic property. This is weaker than the mixing condition of the system needed in the previous estimation strategy.

In formal terms you may introduce the concept of ergodicity as the behavior of a system where the second estimation approach does work. Sadly or luckily, who knows, ergodicity is unlikely to occur in most of the dynamical systems in nature :).