Proof of the Ergodic Theorem.

Birkhoff, G. D. (1931). Proof of the Ergodic Theorem. Proceedings of the National Academy of Sciences USA, 17(12), 656-660.

Type

Journal article

Intellectual base

Primary research

Year

1931

DOI

10.1073/pnas.17.2.656

Link

https://doi.org/10.1073/pnas.17.2.656

Cited from

mathematics, physics

Cited by

3 citations across 3 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Chaos
  • Their unpredictability is individual-trajectory, not ensemble; the long-time average over a single trajectory equals the space-average over the attractor (Birkhoff's (1931) ergodic theorem

    This sourcePointwise ergodic theorem establishing that for measure-preserving transformations time averages along almost every trajectory equal the space average; foundational for treating chaotic dynamics via invariant measures.

• Ensemble
  • . Birkhoff's ergodic theorem

    This sourcethe pointwise ergodic theorem — for measure-preserving transformations, time averages along almost every trajectory equal the space (ensemble) average — justifying ensemble replacement of single trajectories for ergodic systems.

Domain-specific

• Ergodic process
  • A process is ergodic with respect to a specified observable or invariant sigma-algebra when its time average converges, in the stated mode, to a nonrandom ensemble value; full ergodicity makes invariant events trivial.

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