Ergodic Theory

Ergodic Theory is a mathematical theory of random processes in discrete or continuous time. Such a process is called ergodic if all averages over time are the same with probability one, and equal to the expectation value of the quantity considered. A more mathematical definition is: the probability distribution of the process is extremal among the shift invariant ones. The term was coined by Ludwig Boltzmann in the 19-th century to designate a hypothesis which he made about statistical mechanical systems in order to justify the use of equilibrium probability densities. The first ergodic theorems were proved by John von Neumann and George David Birkhoff.