|
|
| Line 1: |
Line 1: |
| {{other uses2|Ergodic}}
| | Hi there, I am Andrew Berryhill. Ohio is where my house is but my spouse desires us to move. To perform lacross is the factor I adore most of all. Office supervising is my occupation.<br><br>my page ... spirit messages ([http://alles-herunterladen.de/excellent-advice-for-picking-the-ideal-hobby/ simply click the up coming site]) |
| | |
| In [[mathematics]], the term '''ergodic''' is used to describe a [[Dynamical system (definition)|dynamical system]] which, broadly speaking, has the same behavior averaged over time as averaged over the space of all the system's states ([[phase space]]). In [[physics]] the term is used to imply that a system satisfies the [[ergodic hypothesis]] of [[thermodynamics]].
| |
| | |
| In [[statistics]], the term describes a random process for which the time average of one sequence of events is the same as the [[ensemble average]]. In other words, for a Markov chain, as one increases the steps, there exists a positive probability measure at step <math> n </math> that is independent of probability distribution at initial step 0 (Feller, 1971, p. 271).<ref>Feller, W. 1971 An introduction to probability theory and its applications, vol. 2, Wiley</ref>
| |
| | |
| ==Etymology==
| |
| The term "ergodic" was derived from the Greek words ''έργον'' (''ergon'': "work") and ''οδός'' (''odos'': "path" or "way"). It was chosen by [[Boltzmann]] while he was working on a problem in statistical mechanics.<ref>{{Harvard citations|last = Walters|year = 1982|loc = §0.1, p. 2|nb = yes}}</ref>
| |
| | |
| ==Formal definition==
| |
| Let <math>(X,\; \Sigma ,\; \mu\,)</math> be a [[probability space]], and <math>T:X \to X</math> be a [[measure-preserving transformation]]. We say that ''T'' is '''ergodic''' with respect to <math>\mu</math> (or alternatively that <math>\mu</math> is ergodic with respect to ''T'') if one of the following equivalent statements is true:<ref>{{Harvard citations|last = Walters|year = 1982|loc = §1.5, p. 27|nb = yes}}</ref>
| |
| | |
| *for every <math> E \in \Sigma</math> with <math>T^{-1}(E)=E\,</math> either <math>\mu(E)=0\,</math> or <math>\mu(E)=1\,</math>.
| |
| *for every <math> E \in \Sigma</math> with <math>\mu(T^{-1}(E)\bigtriangleup E)=0</math> we have <math>\mu(E)=0</math> or <math>\mu(E)=1\,</math> (where <math>\bigtriangleup</math> denotes the [[symmetric difference]]).
| |
| *for every <math> E \in \Sigma</math> with positive measure we have <math>\mu(\cup_{n=1}^\infty T^{-n}E) = 1</math>.
| |
| *for every two sets ''E'' and ''H'' of positive measure, there exists an ''n'' > 0 such that <math>\mu((T^{-n}E)\cap H)>0</math>.
| |
| | |
| === Measurable flows ===
| |
| These definitions have natural analogues for the case of measurable flows and, more generally, measure-preserving semigroup actions. Let {''T''<sup>''t''</sup>} be a measurable flow on (''X'', ''Σ'', ''μ''). An element ''A'' of ''Σ'' is invariant mod 0 under {''T''<sup>''t''</sup>} if
| |
| | |
| : <math>\mu(T^{t}(A)\bigtriangleup A)=0</math>
| |
| | |
| 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''.
| |
| | |
| == Markov chains ==
| |
| In a [[Markov chain]], a state <math>i</math> is said to be ergodic if it is aperiodic and positive recurrent (a recurrent state is one that does not have a 0 transitional probability to exit from it, otherwise it becomes "absorbing"). If all states in a Markov chain are ergodic, then the chain is said to be ergodic. A Markov chain is ergodic if there is a strictly positive probability to pass from any state to any other state in one step (Markov's theorem).
| |
| | |
| == Examples in electronics ==
| |
| Ergodicity is where the ensemble average equals the time average. Each resistor has [[thermal noise]] associated with it and it 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. You should also note that you have ''N'' waveforms as we have ''N'' resistors. These ''N'' plots are known as ensembles. 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. If both 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 [[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.
| |
| | |
| == See also ==
| |
| * [[Measure-preserving dynamical system]]
| |
| * [[Ergodic theory]]
| |
| * [[Mixing (mathematics)]]
| |
| | |
| ==Notes==
| |
| {{reflist|3}}
| |
| | |
| ==References==
| |
| * {{Citation | last1=Walters | first1=Peter | title=An Introduction to Ergodic Theory | publisher=[[Springer Science+Business Media|Springer]] | isbn=0-387-95152-0 | year=1982}}
| |
| * {{Citation | author1=Brin, Michael | author2=Garrett, Stuck | title=Introduction to Dynamical Systems | publisher=Cambridge University Press | year=2002 | isbn=0-521-80841-3}}
| |
| * Birkhoff, G. D. (1931). Proof of the ergodic theorem. Proceedings of the National Academy of Sciences of the United States of America, 17(12), 656.
| |
| * Alaoglu, L., & Birkhoff, G. (1940). General ergodic theorems. The Annals of Mathematics, 41(2), 293-309.
| |
| | |
| ==External links==
| |
| {{wiktionary|ergodic}}
| |
| * [http://www.math.unc.edu/Faculty/petersen/erg3.doc Outline of Ergodic Theory], by Steven Arthur Kalikow
| |
| | |
| [[Category:Ergodic theory]]
| |
| [[Category:Probability theory]]
| |
| [[Category:Stochastic processes]]
| |
| | |
| [[fr:Théorie ergodique]]
| |
| [[pl:Hipoteza ergodyczna]]
| |
Hi there, I am Andrew Berryhill. Ohio is where my house is but my spouse desires us to move. To perform lacross is the factor I adore most of all. Office supervising is my occupation.
my page ... spirit messages (simply click the up coming site)