Ergodicity.

I recently came across a great explainer video on ergodicity by Alex Adamou on Ergodicity.tv

Ergodicity is an incredibly important concept to understand when dealing with multiple trajectories of a stochastic process — which is simply a random process through time. Common examples of stochastic processes include games of chance (i.e. rolling a dice) and random walks (i.e. path a molecule takes as it travels in a liquid or gas).

Taking some examples from Alex, below is a graph representing a stochastic process X(t) with a set of four trajectories {xn(t)}.

To determine the finite time average we fix the trajectory and average horizontally over the time range

When time diverges the time average is given by

In the finite ensemble average we fix time and average over the trajectories

When the number of systems diverge the ensemble average is given by

A stochastic process is ergodic if the time average of a single trajectory is equal to the ensemble average.

Shown graphically, here is what we are examining: