In mathematics and physics, the adjective ergodic is used to imply that a system satisfies the ergodic hypothesis of thermodynamics or that it is a system studied in ergodic theory.
A more rigorous definition may be given as follows:
Let (X,Σ,μ) be a probability space, and T:X→X be a measure-preserving transformation, i.e.
for all 
so μ is an invariant measure under T. We call T an ergodic transformation (with respect to
μ) and call μ an ergodic
measure (with respect to T) if, whenever T(E) =
E for some 
, then
- μ(E) = 0 or μ(E) = 1.
That is, T takes "almost all sets all over the space". The only sets it "doesn't move" are some sets of measure zero and sets that are almost the entire space.
The collection of probability measures on X that are ergodic with respect to T is sometimes denoted ET(X).
External Links
- Outline of Ergodic Theory, by Steven Arthur Kalikow
References
- This article incorporates material from ergodic on PlanetMath, which is licensed under the GFDL.
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