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'''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 occurred. The [[probability distribution]] takes extreme values over the class of shift invariant processes.

'''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 that the end of the time period, the [[random event]] will surely have occurred. 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 [[mechanics]] in a probabilistic model. [[John von Neumann]] and [[George David Birkhoff]] are other seminal contributors to the field.

The first serious study of ergodic theory was performed by [[Ludwig Boltzmann]] in the 1930s, who also coined the term. His thesis was to justify the use of [[equilibria]] to study [[mechanics]] in a probabilistic model. [[John von Neumann]] and [[George David Birkhoff]] are other seminal contributors to the field.

A controversial application of the probability-based '''ergodic theory''' is to [[number theory]], which traditionally has been one of the most rigorous fields of mathematics.

A controversial application of the probability-based '''ergodic theory''' is to [[number theory]], which traditionally has been one of the most rigorous fields of mathematics.

[[category:mathematics]]

[[category:mathematics]]