ergodicity
Sign in to saveIn mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space in which the system moves, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored in
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Encyclopedic overview
46 sectionsContents
- Informal explanation
- Measure-preserving dynamical systems
- Ergodic processes
- History and etymology
- Ergodicity in physics and geometry
- In statistical mechanics
- Simple dynamical systems
- In classical mechanics and geometry
- In wave mechanics
- In quantum mechanics
- Definition for discrete-time systems
- Invariant measure
- Ergodic measure
- Examples
- Equivalent formulations
- Further examples
- Bernoulli shifts and subshifts
- Irrational rotations
- Arnold's cat map
- Ergodic theorems
- Related properties
- Dense orbits
- Mixing
- Proper ergodicity
- Definition for continuous-time dynamical systems
- Examples
- Ergodic flows
- Ergodicity in compact metric spaces
- Functional analysis interpretation
- Existence of ergodic measures
- Ergodic decomposition
- Example
- Continuous systems
- Unique ergodicity
- Probabilistic interpretation: ergodic processes
- Ergodicity of Markov chains
- The dynamical system associated with a Markov chain
- Criterion for ergodicity
- Examples
- Counting measure
- Non-ergodic Markov chains
- A periodic chain
- Generalisations
- Notes
- References
- External links
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space in which the system moves, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity.
Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motions of particles, that is, geodesics, on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the same general area, eventually filling the entire space.
Excerpted from Wikipedia’s “ergodicity” article, available under the CC BY-SA 4.0 licence.