“Ergodicity”的意思、由来-开放百科全书

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Ergodicity

释义

Formal definition
Measurable flows Unique ergodicity
Markov chains
Examples in electronics
Ergodic decomposition
See also
Notes
References
External links

{{Use American English|date = March 2019}}{{Short description|property of a dynamical system that its behavior averaged over time is the same as averaged over its phase space}}{{other uses2|Ergodic}}{{too technical|date=July 2014|I strongly disagree with the view that all subjects must be "dumbed down" to be accessible to the average reader. In this particular case, it is unlikely that the "average reader" would be curious about the concept of ergodicity in the first place. Is there no place for readers who want to be challenged?=}}

In probability theory, an ergodic dynamical system is one that, broadly speaking, has the same behavior averaged over time as averaged over the space of all the system's states in its phase space. In physics the term implies that a system satisfies the ergodic hypothesis of thermodynamics.

A random process is ergodic if its time average is the same as its average over the probability space, known in the field of thermodynamics as its ensemble average. The state of an ergodic process after a long time is nearly independent of its initial state.^[1]

The term "ergodic" was derived from the Greek words {{lang|el|έργον}} (ergon: "work") and {{lang|el|οδός}} (odos: "path", "way"). It was chosen by Ludwig Boltzmann while he was working on a problem in statistical mechanics.^[2] The branch of mathematics that studies ergodic systems is known as ergodic theory.

Formal definition

Let be a probability space, and be a measure-preserving transformation. We say that T is ergodic with respect to (or alternatively that is ergodic with respect to T) if the following equivalent conditions hold:^[3]

for every with either or ;
for every with we have or (where denotes the symmetric difference);
for every with positive measure we have ;
for every two sets E and H of positive measure, there exists an n > 0 such that ;
Every measurable function with is almost surely constant.

Measurable flows

These definitions have natural analogues for the case of measurable flows and, more generally, measure-preserving semigroup actions. Let {T^t} be a measurable flow on (X, Σ, μ). An element A of Σ is invariant mod 0 under {T^t} if

for each t ∈ R. Measurable sets invariant mod 0 under a flow or a semigroup action form the invariant subalgebra of Σ, and the corresponding measure-preserving dynamical system is ergodic if the invariant subalgebra is the trivial σ-algebra consisting of the sets of measure 0 and their complements in X.

Unique ergodicity

A discrete dynamical system , where is a topological space and a continuous map, is said to be uniquely ergodic if there exists a unique -invariant Borel probability measure on . The invariant measure is then necessary ergodic for (otherwise it could be decomposed as a barycenter of two invariant probability measures with disjoint support).

Markov chains

In a Markov chain with a finite state space, a state is said to be ergodic if it is aperiodic and positive-recurrent (a state is recurrent if there is a nonzero probability of exiting the state, and the probability of an eventual return to it is 1; if the former condition is not true, then the state is "absorbing"). If all states in a Markov chain are ergodic, then the chain is said to be ergodic.

Markov's theorem: a Markov chain is ergodic if there is a positive probability to pass from any state to any other state in one step.

For a Markov chain, a simple test for ergodicity is using eigenvalues of its transition matrix. The number 1 is always an eigenvalue. If all other eigenvalues are positive and less than 1, then the Markov chain is ergodic. This follows from the spectral decomposition of a non-symmetric matrix.

Examples in electronics

Ergodicity means the ensemble average equals the time average. Each resistor has an associated thermal noise that depends on the temperature. Take N resistors (N should be very large) and plot the voltage across those resistors for a long period. For each resistor you will have a waveform. Calculate the average value of that waveform; this gives you the time average. Note also that you have N waveforms as we have N resistors. These N plots are known as an ensemble. Now take a particular instant of time in all those plots and find the average value of the voltage. That gives you the ensemble average for each plot. If ensemble average and time average are the same then it is ergodic.

Ergodic decomposition

Conceptually, ergodicity of a dynamical system is a certain irreducibility property, akin to the notions of irreducibility in the theory of Markov chains, irreducible representation in algebra and prime number in arithmetic. A general measure-preserving transformation or flow on a Lebesgue space admits a canonical decomposition into its ergodic components, each of which is ergodic.

Notes

1. ^{{cite book |first=William |last=Feller |title=An Introduction to Probability Theory and Its Applications |edition=2nd |url={{google books |plainurl=y |id=OXkg-LvRgjUC |page=271}} |date=1 August 2008 |publisher=Wiley India Pvt. Limited |isbn=978-81-265-1806-7 |page=271}}
2. ^{{Harvard citations|last = Walters|year = 1982|loc = §0.1, p. 2|nb = yes}}.
3. ^{{Harvard citations|last = Walters|year = 1982|loc = §1.5, p. 27|nb = yes}}

References

{{Cite book | last1=Walters | first1=Peter | title=An Introduction to Ergodic Theory | publisher=Springer | isbn=0-387-95152-0 | year=1982}}
{{Cite book | author1=Brin, Michael | author2=Garrett, Stuck | title=Introduction to Dynamical Systems | publisher=Cambridge University Press | year=2002 | isbn=0-521-80841-3}}
{{cite journal |last=Birkhoff |first=George D. |authorlink=George David Birkhoff |date=December 1931 |title=Proof of the ergodic theorem |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=17 |issue=12 |page=656 |doi=10.1073/pnas.17.2.656 |doi-access=free |pmc=1076138 |jstor=86016 |url=http://www.pnas.org/content/17/12/656.full.pdf |pmid=16577406}}
{{cite journal |last=Alaoglu |first=Leonidas |authorlink1=Leonidas Alaoglu |last2=Birkhoff |first2=Garrett |authorlink2=Garrett Birkhoff |date=April 1940 |title=General ergodic theorems |journal=The Annals of Mathematics |series=2 |volume=41 |issue=2 |pages=293–309 |jstor=1969004 |mr=0002026 |doi=10.2307/1969004|pmc=1077986 }}
{{cite journal |last=Alaoglu |first=Leonidas |authorlink1=Leonidas Alaoglu |last2=Birkhoff |first2=Garrett |authorlink2=Garrett Birkhoff |date=December 1939 |title=General ergodic theorems |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=25 |issue=12 |pages=628–630 |pmid=16588311 |pmc=1077986 |jstor=87048 |doi=10.1073/pnas.25.12.628 |doi-access=free |url=http://www.pnas.org/content/25/12/628.full.pdf}}

External links

Steven Arthur Kalikow, "Outline of Ergodic Theory"
Karma Dajani and Sjoerd Dirksin, "A Simple Introduction to Ergodic Theory"

Théorie ergodiqueHipoteza ergodyczna

1 : Ergodic theory

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