ergodic theory

Ergodic theory is a branch of mathematics that deals with systems where every sample is equally representative of the whole. It is particularly concerned with the probability that any state within a system will recur.1

A key concept is the ergodic theorem, first rigorously formulated by George David Birkhoff in 1931. This theorem states that, under certain conditions, the time average of a system's measurements over an extended period will equal the space average of those measurements taken at a single point in time. For example, the average temperature of a room measured over time is equivalent to the average temperature measured simultaneously across all points in the room. This principle has found applications in various fields, including statistical mechanics and number theory.23