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'''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]].

'''Ergodic Theory''' is the [[mathematical]] theory of well-behaved [[random]]ness. The [[random variable]]s can occur [[discrete]]ly or [[continuous]]ly, but the total [[probability]] over the time period is always 1 (so by the end of the time period, the [[random event]] will surely have occured. The [[probability distribution]] takes extreme values over the class of shift invariant processes.

 

The first serious study of ergodic theory was performed by [[Ludwig Boltzmann]], who also coined the term. His thesis was to justify the use of [[equilibria]] to study [[mechanic]]s in a probabilistic model. [[John von Neumann]] and [[George David Birkhoff]] are other seminal contributors to the field.

[[category:mathematics]]

[[category:mathematics]]