Talk:Ergodicity

message: It would be good to list some simple implications of ergodicity.  For example, if a process is ergodic, does that imply that it is stationary?  Does it imply that time averages are equal to ensemble averages?
Similarly, it would be good to state something about ergodicity in simple systems, like stating the conditions for a Markov chain to be ergodic. 131.215.45.226 (talk) 19:39, 29 June 2008 (UTC)
thread_title:  Implications of ergodicity 

message: For non-physicists I've introduced an example about call-centre operators, based very closely on the resistor example.  
What is still needed is a discussion of when this would be non-ergodic.  
For instance, would the following be examples of ergodicity or non-ergodicity?
*Refreshments are served hourly, so that workers tend to take their break around the same time, i.e. on the hour.
*Some operators are chatterboxes, and interrupt the person calling in, whereas other operators are more taciturn, or more reluctant to interrupt the person calling in, or better listeners.  The first type of call centre worker will have a much higher average number of words spoken per minute than the latter type.  
*More people call in with enquiries during lunchtime.  
It is also a bit unclear in the wording of the examples as to whether the time average is for one person/resistor, or for the whole group.
And it is unclear why only one ensemble average is captured (at one point in time) to test for ergodicity.  Surely in practice many timepoints should be chosen for (separate) testing.  
Finally, I am not entirely keen on the word "waveform" to describe random noise.  A 'wave' connotes (oftentimes smooth) periodic behaviour.
—DIV (137.111.13.48 (talk) 02:50, 24 April 2019 (UTC)
replies: text: I am agreement with the comments above. The call centre example needs to be re-written and clarified. Oldtimermath (talk) 17:56, 18 September 2019 (UTC)
thread_title: Call centre example

message: I found a good reference online that I believe should be incorporated into either or both Ergodicity and Stationary processes. In general, it appears as if the major difference is that ergodicity has to do with "asymptotic independence," while stationary processes have to do "time invariance":
http://economia.unipv.it/pagp/pagine_personali/erossi/rossi_intro_stochastic_process_PhD.pdf
However, the above resource has some problems. It introduces a process that it claims is stationary, but not ergodic, and proceeds to prove the process is stationary. Then the process is redefined as a random walk and proved it is not ergodic. However the random walk is not stationary as the variance grows linearly in time. Are there any better examples that we could use to demonstrate (1) a process that is ergodic but not stationary and (2) a process that is not ergodic, but is stationary?  — Preceding unsigned comment added by 150.135.222.130 (talk) 22:59, 31 January 2013 (UTC)
thread_title:  Clarification of Ergodicity vs Stationary 

message: The external source file, Outline of Ergodic Theory, by Steven Arthur Kalikow, is listed as a Word document. Please reupload as a .pdf file. This is much more convenient.  — Preceding unsigned comment added by 69.77.224.241 (talk) 22:31, 17 August 2013 (UTC)
replies: text: You'd have to ask the author of that paper to do that; as a copyrighted document it must be hosted externally. -- Beland (talk) 14:28, 11 July 2014 (UTC)
thread_title: External Source Improvement

message: A queue is an example of a continuous-time Markov chain. A queue has three components - inter-arrival times, server times and the number of servers; the first two of these may be Markov processes.
Assuming the inter-arrival time is a Markov process, then any particular state is recurrent (or persistent) if the probability of ever returning to it is 1 (i.e. certain).
Hence if between 2 states the time is 30 seconds (the time between two entities joining the queue) then this indicates that the second state is recurrent (or persistent) because this situation could occur again.
If the expected time between the two states is finite (as opposed to infinite) then the second state is ergodic. 
Hence the time between 2 people joining a queue is finite, could occur again and is hence ergodic.Oxford Dictionary of Statistics,   Neugierigxl (talk) 14:40, 10 May 2015 (UTC)
thread_title: Queues

message: There is a formal definition without any examples. Some simple examples and counter examples would really help the reader and seem necessary.
31.39.233.46 (talk) 18:22, 25 April 2016 (UTC)
thread_title:  Lack of examples 

message: I propose to merge Ergodicity into Ergodic process
77.81.10.206 (talk) 20:29, 29 December 2018 (UTC)
replies: text: Kill me if you will, or revert, but even before noting there was a merge request in progress, I basically rewrote parts of the article in a way supportive of a merge. If anything, that sort of thing makes me think a merge would be worthwhile -- if it comes out of sheer instinct, I tend to rely on it. Decoy (talk) 22:40, 13 March 2020 (UTC)
thread_title:  Merger proposal 

message: Is there a way to interpret ergodicity in Markov chains as a special case of the Dynamical systems definition? It seems as though ergodicity in dynamical systems is more closely related to the irreducibility of Markov chains as mentioned in the Ergodic decomposition section. If they are fundamentally different, then this should be explicitly mentioned in the Markov chain section. On the other hand, if there is some way to connect the two, then that should be mentioned
: There is no fundamental difference. In dynamical systems you use a continuous measure over the phase space. With Markov chains, you use a discrete one (often finite as well). That irreducibility stuff mostly works out the same as well, except for the complications having to do with limiting arguments, weak topologies, and whathaveyou. But as I said, no big brouhaha; in fact this is why we start out with the measure theoretic definition, which already bridges the two scenarios. Decoy (talk) 22:45, 13 March 2020 (UTC)
replies: text: Unfortunately, under the typical way of viewing a Markov chain as a DS, they are not the same, and it is the case that "ergodic" for Markov chains becomes "mixing" in the general context of DS.  And you are right that "irreducible" for Markov chains is "ergodic" for DS.  To view a Markov chain with state space S={1,...,n} as a dynamical system, you consider the phase space X consisting of all infinite paths through the phase space, like (a_1,a_2,a_3,...), and your map T shifts everything left by 1: T(a_1,a_2,...)=(a_2,a_3,...).  This is often called a shift map.  And finally you equip the space with a measure mu which is consistent with the transition probabilities of the Markov chain (also you can check that in order for T to be measure-preserving, you need mu(a x S x S x S...) to equal a stationary distribution P(a) for the Markov chain). 192.16.204.209 (talk) 22:17, 13 April 2022 (UTC)
thread_title:  Ergodicity in Markov chains vs Dynamical systems 

message: There are systems that are dense but not ergodic; see, for example MathOverflow: Example of a measure-preserving system with dense orbits that is not ergodic but the example is too complex for this article. There is another example to be found in translation surface, which states "there may be directions in which the flow is minimal (meaning every orbit is dense in the surface) but not ergodic". Based on this, I suspect there are no simpler examples, but hope never dies. 
There's also this fascination discussion on ergodicity and dense orbits.
Searching for "minimal but not uniquely ergodic" has many hits, including:
* Alistair Windsor Minimal but not uniquely ergodic diffeomorphisms
* The Jewett-Krieger Theorem, quoting Uniquely ergodic and strongly mixing transformation I see this jewel: The Jewett-Krieger Theorem states that every ergodic measure-preserving transformation of a standard probability space can be realised as a uniquely ergodic topological dynamical system on a compact metric space. If the metric entropy of the original system is strictly less than \log d, where d is an integer, then additionally this topological dynamical system can be taken to be a closed shift-invariant subset of the full shift on d symbols. A proof of this theorem can be found in the book Ergodic theory on compact spaces by Denker, Grillenberger and Sigmund. There's also stuff about horocycle flows being uniquely ergodic and mixing.
But this is walking into a bottomless pit: there's an ocean of interesting stuff to be said about ergodicity, and what this article currently says is the tip of the iceberg... 67.198.37.16 (talk) 05:41, 5 November 2020 (UTC)
thread_title:  Dense but not ergodic 

message: This page discusses the concept of ergodicity in pure mathematical terms. There's nothing wrong with this, but I suspect many people searching for "Ergodicity" are interested in the article on ergodic processes. I propose renaming the articles; this article should be renamed "Ergodicity (Mathematics)." "Ergodic process" should be renamed "Ergodicity (statistics)," and the two articles should be clearly disambiguated with a "Not to be confused with" label.
Closed Limelike Curves (talk) 01:16, 3 April 2022 (UTC)
replies: text: This page describes ergodic processes as well, and attempts to show how the dynamical-systems version and the stochastic process version are the same. However, ergodic process is oddly unlinked in this article; I'll try to fix that shortly. 67.198.37.16 (talk) 14:07, 20 April 2023 (UTC)
thread_title:  Rename and disambiguate?