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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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.