ergodic theorem

The ergodic theorem, first rigorously proven by George David Birkhoff in 1931, is a fundamental concept in probability theory and statistics. It states that for certain systems, the average behavior over a long period of time (time average) will converge to the average behavior across all possible states at a single point in time (space average).1

For example, imagine measuring the temperature of a room at regular intervals over several hours. This is the time average. If you could simultaneously measure the temperature at every single point in the room and average those readings, that would be the space average. The ergodic theorem suggests that, under specific conditions, these two averages will become equal as the number of measurements increases indefinitely. This principle has broad applications, including in statistical mechanics and number theory.12