Thankfully, this blog is no longer the number one hit for “ergodicity” on Google. But in the event that people come here anyway, I present some definitions, courtesy of Wikipedia, Richard Durrett, and others.
The ergodic hypothesis says that averaging over time and averaging over the statistical ensemble are the same. So let’s say I have some box spitting a random number every second. If the random process controlling the box is ergodic, then I can find certain quantities — for example, the average — by either averaging the observed variables that I see, or by calculating the “theoretical” average from the statistics governing the box.
We would, of course, like most real-world systems to be ergodic, since we can then measure them and make estimates based on the measurements. The hope is that these estimates will (in the limit as you get infinite data) converge to the “real” value. Of course, this leads to an existential bind, because we have no idea if there is a “real” underlying value.
It’s a tricky thing, ergodicity, and getting to the bottom of it reveals a lot about how we view the randomness in our world, the assumptions we make on it, and how we try to control it.