Birkhoff's ergodic theorem

Birkhoff's ergodic theorem states that in a system where states evolve over time according to certain rules, the long-term average of a property (like energy or position) along a single trajectory will match the overall average across all possible states, assuming the system is "ergodic." This means that, over a long period, observing one typical path of the system gives a representative picture of its entire behavior. Essentially, the theorem links long-term time averages with average behavior across all states, ensuring consistent statistical predictions in such dynamic systems.