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In physics, statistics, econometrics and signal processing, a stochastic process is said to be in an ergodic regime if an observable's ensemble average equals the time average. [1] In this regime, any collection of random samples from a process must represent the average statistical properties of the entire regime.
The mathematical definition of ergodicity aims to capture ordinary every-day ideas about randomness.This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as diffusion and Brownian motion, as well as common-sense notions of mixing, such as mixing paints, drinks, cooking ingredients, industrial process mixing, smoke in a smoke-filled room, the ...
Ergodic theory is often concerned with ergodic transformations.The intuition behind such transformations, which act on a given set, is that they do a thorough job "stirring" the elements of that set. E.g. if the set is a quantity of hot oatmeal in a bowl, and if a spoonful of syrup is dropped into the bowl, then iterations of the inverse of an ergodic transformation of the oatmeal will not ...
In probability theory, a stationary ergodic process is a stochastic process which exhibits both stationarity and ergodicity.In essence this implies that the random process will not change its statistical properties with time and that its statistical properties (such as the theoretical mean and variance of the process) can be deduced from a single, sufficiently long sample (realization) of the ...
Assumption of the ergodic hypothesis allows proof that certain types of perpetual motion machines of the second kind are impossible. Systems that are ergodic are said to have the property of ergodicity; a broad range of systems in geometry, physics, and probability are ergodic. Ergodic systems are studied in ergodic theory.
Bernoulli shifts are classified by their metric entropy. [ 11 ] [ 12 ] [ 13 ] See Ornstein theory for more. Krieger finite generator theorem [ 14 ] (Krieger 1970) — Given a dynamical system on a Lebesgue space of measure 1, where T {\textstyle T} is invertible, measure preserving, and ergodic.
Aside from its generic use as the generic adjective ergodic, ergodic may relate to: Ergodicity, mathematical description of a dynamical system which, broadly speaking, has the same behavior averaged over time as averaged over the space of all the system's states (phase space) Ergodic hypothesis, a postulate of thermodynamics
In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces.Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subgroup of the Möbius group G = PSL(2,R), then the geodesic and horocycle flow ...