Talk:Ergodicity

Implications of ergodicity
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)

Hurst exponents
Similarly, it would perhaps be interesting to explore relations between ergodicity and Hurst Exponents for specific time series. Is the Hurst Exponent (one of) the invariant(s) of an ergodic process?

Expert help: intuitive definition
The section "Intuitive definition" seems wrong and misleading. But, rather than deletion, can something better be said? Melcombe (talk) 12:49, 25 June 2010 (UTC)
 * In fact, on re-reading, the whole article seems confused. Melcombe (talk) 12:53, 25 June 2010 (UTC)

Even the math definition is insufficient. For instance, what does T^^-n mean? That T should be inverted n times? What is the point of such a definition? — Preceding unsigned comment added by 212.27.17.76 (talk) 09:39, 26 October 2019 (UTC)


 * The article is certainly confusing and evidently written by a pure mathematician in a style which will be completely unintelligible to most people who want to understand the idea. The term ergodicity is understood in different ways in (1) physics (2) pure mathematics (3) statistics and systems analysis. The article should pay attention to each meaning and not concentrate on just one interpretation (here the pure mathematical one).  The interpretations are of course related and the relation can be understood from the historical development.JFB80 (talk) 20:40, 21 April 2011 (UTC)

I think it would help physicists to give at least a sketch of a concrete example. For example, for a classical point-particle moving in a potential: X corresponds to phase space, Σ corresponds to the Lebesgue measurable subsets of X, and υ is the Lebesgue measure. In particular, I was initially thinking that υ was a physical probability distribution on X (e.g., it might be strongly concentrated around a particular point x_1 following a measurement) rather than a measure that's proportional to density of microstates. 174.28.23.23 (talk) 07:44, 8 December 2011 (UTC)


 * I agree a better intuitive description is needed; a visual representation would also help a lot. -- Beland (talk) 14:29, 11 July 2014 (UTC)


 * I add my support for inclusion of examples for two cases with some graphical representations.
 * One corresponding to the 'physics' definition (maybe maps of the probability of a particle visiting some regions??).
 * One corresponding to the 'statistics' definition (a plot of some random variable changing over time).
 * In both cases there should be contrasting examples illustrating ergodicity and lack of ergodicity so that we can see the difference.
 * —DIV (137.111.13.4 (talk) 00:55, 19 March 2015 (UTC))
 * Agreed. And the boundary would be clearer with some examples of systems that are, only just, not ergodic. JDAWiseman (talk) 09:25, 14 June 2018 (UTC)

Call centre example
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? 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))
 * 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.
 * 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)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: page moved per request. - GTBacchus(talk) 02:03, 20 September 2010 (UTC)

Ergodic (adjective) → Ergodicity — Current name is unsatisfactory (the adjective part). Ergodicity, which is the noun form, seems preferable. Tiled (talk) 00:50, 12 September 2010 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Clarification of Ergodicity vs Stationary
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)

External Source Improvement
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)
 * 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)

Example from electronics
This section is concerned with the thermal or Johnson noise exhibited by collections of resistors. As a physical phenomenon, thermal noise is the low frequency band of black body radiation, and as such at ν = 0 the emission according to the set of temperature curves (Wein's law) for the blackbody process all approach zero at that frequency. Therefore the average voltage measured must be always zero because the average voltage is the D.C. quantity, and D.C. is the spectral output at f = 0 Hz using the symbol from electrical engineering. BTW v should be lower case Nu in the first equation, which does not correctly appear.

The section maybe could mention whether the measurements should all be at the same temperature T or not, and whether or not all of the resistors are of identical value R. Measuring this from resistors of identical value R and temperature T are analogous to measuring/determining radiation output of black body objects of identical surface area and identical T. If the resistors are not of identical R and T, then measuring each resistor Johnson noise output can be an indirect way of determining R of each or the T of each if one or the other is known.

And instead of the measurement across the resistors being of the D.C or average voltage, or the same thing calculated from instantaneous samples of the voltages, the measurements should be of the R.M.S. voltage of the resistor thermal voltage or RMS volts v = σ. But even specifying this presents a problem in that the RMS quantity has a specification of bandwidth, as the RMS quantity represents the integral of the noise density spectrum (in volts/Hz^1/2) over df. And if all resistors are of identical R then by the temperature curves the RMS voltage values of each are identical, taken over identical bandwidth.

BTW the RMS as the integral over df is calculated using density spectrum in volts/(Hz^1/2); the reason is that each df individual σ cannot be added linearly unless they are all cross-correlated r = 1. But with ergodic processes any two df are cross-correlated r = 0, and so the voltages over the set {df} are added non-linearly as RMS which is calculated as the integral over df.

I do not know if any of my concerns can be incorporated into the article as the statistical nature of what is proposed possibly does not warrant the details from this post. But I think the section can be improved/clarified based on what I'm putting here.

Groovamos (talk) 10:16, 10 March 2014 (UTC)


 * The section "Example from electronics" is pretty bad. It tediously re-explains the concept of ergodicity without adding anything specific from electronics. It's empty with choppy grammar. After reading it, I have no idea why it's interesting or useful to compare time and space averages of a very large number of unrelated resistors.


 * By the way, Groovamos, how do you know all those things that you write about the significance of ergodicity in electronics? I don't understand them, but it suggests to me that ergodicity in electronics really is a thing. If there were a source, it would help greatly. I couldn't find the word "ergodicity" in the article on Johnson–Nyquist_noise.


 * 178.39.122.125 (talk) 02:28, 9 January 2017 (UTC)

Actually I have never had to pay attention to the formal categorizations of noise in dealing with noise in electronic design. I first encountered the categorizations of stochastic processes in the book "Methods of Signal and System Analysis" and have never really understood very well the implications for system design of ergodic noise sources. But I have never had to work with deep space telecommunications either so that explains my first sentence. You are right about the section being somewhat disjointed but it has actually helped me in my old age to understand something better, and that is that a system with a single or dominant source of noise can possibly be analysed without regard to ergodicity. But I'm now seeing something decades later in that the methods in that book are based on the multiple sources of noise usually present in a system, and some of the techniques apply to obtaining an accurate insight into system behavior by for example obtaining the stochastic output due to all the ergodic sources as an ensemble of sources and then treating the non-ergodic sources independently and combining by superposition. And my reference to black body radiation as being the same physical process as Johnson noise means that the contributions of noise in the microwave region at a receiving antenna can be more easily accounted for in a system when taking into account at the same time the Johnson noise added by the resistive components in a system and treating them as an ensemble of noise sources to get the system stochastic response.Groovamos (talk) 21:44, 6 June 2017 (UTC)


 * In the example it is unclear whether time-averages for individual resistors need to be equal to one another. —DIV (137.111.13.48 (talk) 02:24, 24 April 2019 (UTC))

Queues
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. Neugierigxl (talk) 14:40, 10 May 2015 (UTC)

Boolean Networks
A Boolean network is said to be ergodic if it cycles through all possible states of the network, visiting each state only once and returning to its initial state.

Lack of examples
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)

Dark matter
If a quantum phenomenon occurs at a smaller pace (if compared to a more dense region of space), that phenomenon generates gravity - if compared to it's surroundings - in order ergodicity is maintained.

Problem with Markow chain
Hi everybody, I have a problem with what is written on the paragraph on Markov chains: are we sure that if all eigenvalues are smaller than 1 then the matrix is ergodic?? I cannot find any reference for it. Plus consider the example:

This matrix is clearly not ergodic, even if the eigenvalues are (1,0,0,0).

Am I missing something?

Arsik87 (talk) 17:31, 14 June 2018 (UTC)


 * Yes you are missing something. The article says one eigenvalue must be 1 and all the others less than 1 in absolute value. This is not so with your matrix. There is no probability in your example and it is not a proper Markov chain. JFB80 (talk) 10:28, 15 June 2018 (UTC)


 * "The article says one eigenvalue must be 1 and all the others less than 1 in absolute value. This is not so with your matrix." Yes it is since the eigenvalues are less than 1.
 * " There is no probability in your example and it is not a proper Markov chain" I am sorry what do you mean it is not a Markov chain? It is a transition matrix of a Markov chain. If you mean some conditions on the probabilities, then the theorem in the section should specify that.


 * Moreover,I think that also the reported theorem is not totally correct, since the requirement to go from one state to any other in one step is not a necessary condition, e.g. the transition matrix:

is ergodic, but you cannot go from one state to any other state in "one" step.Arsik87 (talk) 23:03, 17 June 2018 (UTC)
 * This matrix is cyclic not ergodic. You don't seem to understand what ergodicity means. It means there is a stationary probability distribution which is attained from any starting state. You quote some books. What do they say? Why not check up yourself? JFB80 (talk) 04:51, 18 June 2018 (UTC)


 * I didn't quote any book and your attitude towards me was not nice from the first comment. Anyway. I pointed out that something in the page might be not as clear as it is for an "expert" as you. The scope of wikipedia should be to explain stuff also to "stupid" people like me.
 * In the text, it says "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 less than 1 in absolute value, then the Markov chain is ergodic. "
 * I m just asking what are the assumptions behind this statement, as my first transition matrix has eigenvalues less than 1 but is not ergodic.
 * Regarding the second matrix, the stationary distribution is (1/4,1/4,1/4,1/4), which is attained from any starting state.
 * ps. If I didn't understand what ergodicity means, isn't it a sign that the section can be improved? Arsik87 (talk) 11:37, 18 June 2018 (UTC)
 * I have said the eigenvalues must be positive and hope it is ok now. I am not surprised you do not know what ergodicity means from reading this article. Ergodicity as a general idea is a very difficult subject and this article and the others on ergodicity need a lot of improvement. But on Markov processes you can find good accounts and some talk about ergodicity. The result mentioned in the article is quite obvious if you understand what spectral resolution of a matrix means. Sorry to upset you. JFB80 (talk) 10:40, 19 June 2018 (UTC)

Merger proposal
I propose to merge Ergodicity into Ergodic process 77.81.10.206 (talk) 20:29, 29 December 2018 (UTC)

I have added the relevant template to the article for this. —DIV (137.111.13.48 (talk) 02:39, 24 April 2019 (UTC))


 * Neutral. This could be a good idea, but please provide more detail on why and how,  What advantages this would have, and would there be any disadvantages? —DIV (137.111.13.48 (talk) 02:27, 24 April 2019 (UTC))


 * Neutral. I think many people who want to read one page should also be interested in the other.  However, perhaps the "See also" link is sufficient?  Servalo (talk) 06:15, 19 May 2019 (UTC)


 * I'd argue the merge with Ergodic Process should indeed happen. That's because the whole idea of ergodicity has no content absent an underlying stochastic process, proceeding in time. It's impossible to talk about ergodicity without a process definining the ensemble, and a time dimension which might somehow make the process unfold differently with starting conditions of said process. So the two concepts go rather tightly hand in hand, with neither giving much more to the analysis than the other.


 * 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)


 * I agree. Merging is the thing to do. Doctormatt (talk) 04:55, 14 March 2020 (UTC)


 * (Strongly) Oppose. There are plenty of things called "ergodic" that are not "processes". The idea of a "process" comes from probability theory and statistics. However, common applictions in mathematics are from the point of view of measure theory.  For example, orbits on Riemann surfaces are ergodic; you'd have to do some torturous manipulations to convert these into a statistical "process" of some kind. Well, that might actually be educational :-) but it would be highly non-standard. Another example: parts of systolic geometry are about ergodicity; its very much not about "processes". The Axiom A and the Hopf fibration and the Anosov diffeomorphism are all ergodic and they are certainly not processes in any conventional sense. Considerable attention in Riemannian geometry is paid to fibrations which decompose the manifold into hyperbolic sets with stable manifolds and unstable manifolds which are assigned a volume entropy ... you'd be hard-pressed to go and rewrite all of these geometries as some kind of statistical processes, and convert the proofs of ergodicity into proofs about statistical processes.  What the heck. Lets throw in Ornstein isomorphism theorem which classifies the different kinds of ergodic systems, and also mention Markov odometer for good measure. Have fun with that :-) 67.198.37.16 (talk) 19:43, 3 November 2020 (UTC)

Ergodicity in Markov chains vs Dynamical systems
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)


 * 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)

Non-ergodicity example
For the 000... and 111... example, isn't the function T(000...) = 111... and T(111...) = 000... ergodic? The first property seems to hold. --Davyker (talk) 20:07, 14 May 2020 (UTC)


 * The discussion concerns the non-ergodicity of the specific Markov chain on $$\{0, 1\}$$ which stays in the same state a.s. ; the transformation you define has nothing to do with it.


 * The article is a bit confusing because it discusses ergodicity for measure-preserving transformations and for finite Markov chains without making a link between the two (also the section on Markov chains is a mess). jraimbau (talk) 15:11, 15 May 2020 (UTC)

Rewrite May 2020
I rewrote the article from the viewpoint of dynamical systems, which seems to me to be the correct one in this particular place. While writing I did not try to address all comments above pointing out the impenetrability; rather, I think that people interested in a less formal definition should go to ergodic process or ergodic hypothesis which deal with particular facets the topic from different viewpoints which are certainly more suitable for people with a natural sciences or information theoretic background. I tried to make that clear in the lead paragraph, and also in the less formal discussion before the formal definition

I moved some examples which seemed a bit belabored but potentially interesting to the ergodic process where I think they are more useful; on the other hand I added some formal but extremely simple examples for people who want to grasp the notion from a mathematical viewpoint, which seems to me to be the goal of this article.

I gave the short end of the stick to continuous-time systems and this should be remedied to at some point (this is a reflection of the fact that I used Walters' book as a main reference, which is excellent and rather complete for ergodic transformations but does not deal with continuous systems). There are plenty of other things that should be added and places where the current writing is suboptimal but I think the article is now a good starting point. jraimbau (talk) 08:17, 18 May 2020 (UTC)


 * I don't know what the article looked like before, but to an outsider--me--it is now completely incomprehensible. Even the examples are simply mathematical, with some notation I'm not familiar with.  (For example, what is the funny-looking N?)  Reading other parts of the Talk Page, I see that there used to be examples from electronics and call centers, and an intuitive definition.  There may have been problems with these, but they surely helped to put some flesh on the mathematical bones.  As it stands, it sounds like ergodicity is a concept only in pure math, whereas I'm told there are real examples (including biology). Mcswell (talk) 02:28, 24 August 2020 (UTC)
 * Replying (as it were) to myself: it looks like the intuitive definition and examples are in the article on Ergodic Processes. I guess that article was the explanation I was looking for.  I do hope if these two articles are combined, that the intuitive defn and examples are not lost. Mcswell (talk) 02:31, 24 August 2020 (UTC)


 * I just now added a long informal discussion which hopefully makes things understandable. 67.198.37.16 (talk) 21:16, 3 November 2020 (UTC)


 * As to examples, there seem to be many, many many people who think ergodicity is about ergodic processes, which could not be farther from the truth. Thus, we need more examples, and maybe those examples should come before, not after the formal definition. This includes: Axiom A and Hopf fibration and Anosov diffeomorphism and orbits in Riemann surfaces and generally geodesics in Riemannian manifolds and the fibrations which decompose the manifold into hyperbolic sets with stable manifolds and unstable manifolds which are assigned a volume entropy. The Hopf bifurcation which is what drives the ergodicity. There's also systolic geometry.  There should also be appeals to generic results: the Ornstein isomorphism theorem which classifies the different kinds of ergodic systems, the various anti-classsification results, and the Markov odometer, which states that any topologically ergodic system is isomorphic to the odometer (aka "adding machine").  Only after these examples and "mind-blowing" results should we dive into the tedium of formal definition. 67.198.37.16 (talk) 21:28, 3 November 2020 (UTC)


 * I've spewed this word-salad of jargon into the article. Perhaps it's readable and understandable. 67.198.37.16 (talk) 00:42, 5 November 2020 (UTC)


 * I trimmed the section on geometric examples, i think the idea of "a quick chain of examples" is good but this section was muddled and had some wrong statements. In its current version it is about flows on riemannian manifolds, which as far as i can tell were the only actual examples in the previous version, the rest being a lot of jargon thrown around. I also tried to give an informal description of the geodesic flow on a flat torus.


 * As a further note: I removed the example of irrational rotations which was given here, it is a good example but does not particularly belong in this section. It was already mentioned in the latter sections ; perhaps a more informal introduction could be given in a nother subsection in this part of the article, maybe together with Bernoulli shifts which are another fundamental example. jraimbau (talk) 14:43, 20 December 2021 (UTC)

Dense but not ergodic
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)

Rename and disambiguate?
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)


 * 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)

Definition of ergodicity conflated with mixing in Informal Explanation
The definition of ergodicity in the Informal Explanation section is currently given as

If a set $$A\in\mathcal{A}$$ eventually comes to fill all of $$X$$ over a long period of time (that is, if $$T^n(A)$$ approaches all of $$X$$ for large $$n$$), the system is said to be ergodic.

But this is a description of mixing, not ergodicity. $$T^n(A)$$ does not need to approach $$X$$, as in the case of an irrational rotation of the circle.

An informal definition of ergodicity is needed which does not conflate with mixing, which is a stronger property. — Preceding unsigned comment added by 192.16.204.209 (talk) 21:50, 13 April 2022 (UTC)


 * The definition for mixing (mathematics), and for ergodicity, are quite similar, but ever so slightly different, if you look carefully. Mixing is described by $$A,B\in\mathcal{A}$$ such that $$A$$ and $$B$$ approach each other over a long period of time, for any A and B. Ergodicity is that same, but only for the one specific B=X being the grand-total space.


 * There is an analogous statement for 3-mixing, involving $$A,B,C\in\mathcal{A}$$, and also for N-mixing for any N. Note that 1-mixing aka "ergodicity" does not imply 2-mixing (aka ordinary (weak) mixing)! There are systems that are ergodic but do not mix. It is not yet known if 2-mixing implies 3-mixing! There are many astounding results of this kind. 67.198.37.16 (talk) 14:04, 20 April 2023 (UTC)

Recent edits
I restored the older version---before the dubious "in geometry" section was restored to its previous execrable version. This section is extremely muddled, the statemens in there are vague to the point of being useless. The current bare-bones version is much better. I suspect the same is true of the other sections in this introductory part but lacking expertise in these domains i did not touch them. jraimbau (talk) 04:58, 20 April 2023 (UTC)


 * Perhaps you could indicate those parts that are muddled or vague or feel "useless". I will re-write or expand to cover the issues. My goal is to make this topic accessible even to beginners (although the "in geometry" section is a bit beyond the beginner level, and assumes some knowledge of geometry.) 67.198.37.16 (talk) 13:48, 20 April 2023 (UTC)
 * Before i cut into it this section was little more than an amalgamation of keywords vaguely related to the topic, with little relevance to the precise topic of the page. The present version is concise and includes only those examples that are really germane to ergodicity (not the general theory of flows, or general ergodic theory---there are separate pages on these topics). As such it seems a much better starting point for a beginner. It might not be super well-written and you are welcome to improve it but it would be counter-productive to expand it much. jraimbau (talk) 13:55, 20 April 2023 (UTC)


 * Hmm. Which keywords are vaguely related to the topic? Almost all theorems about ergodicity are stated in terms of flows on manifolds; I guess the exception would be the stuff that the economics and stochastic process people have obtained (I do not follow those topics). Although, FYI, there is the idea of information geometry which attempts to unify the two approaches. I am not aware of any WP pages on "general ergodic theory". Can you link these? Anyway, talking about flows on low-dimensional manifolds is central to the topic. Elucidating this was one of the grand achievements of the 1980's and 1990's (and has fascinating links to number theory, where it remains a topic of research to this day, e.g. the work of Terrence Tao on thin groups and arithmetic progressions.) 67.198.37.16 (talk) 14:38, 20 April 2023 (UTC)
 * You clearly have no clear idea about what you are talking about, or about what encyclopedical content means in the context of an article on a rather narrow topic. I think this discussion would benefit from a larger audience, I'll put it up on https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics. jraimbau (talk) 07:12, 21 April 2023 (UTC)


 * OK. Except I do know what I'm talking about, And you started a conversation with an insult. Look, we live in the age of search engines. College professors post class notes on ergodicity on-line. There are dozens of PDF's out there. Download them. Read them. Educate yourself. Once you find some evidence that I'm wrong about something, and are able to articulate what it is, then sure ... there's plenty of subtlety and fine points that are hard to capture in such a short article.


 * As to encyclopedic content, well ... there's a way to understand that, too. Go out and talk, face-to-face, with actual people who are actually trying to learn mathematics from Wikipedia. I have. The consistent complaint that I hear is that they are frustrated with the wall of formulas, lacking in any sort of explanation or intuitive insight into what is "actually happening" with those formulas. What I am trying to do here, as with all the other edits I've made, is to explain complex mathematical topics to non-experts, using plain and simple sentences, plain and simple verbal explanations. Now, of course, if someone wants the actual precise and exact definition of something, then yes, they'd have to stare at the formulas and ponder what they actually mean. But for those who only want a general survey, an overview, then a more informal introduction is exactly what is needed. This is what we should strive for: an encyclopedia that non-experts find informative, and experts find useful. 67.198.37.16 (talk) 20:20, 21 April 2023 (UTC)

The conversation continues at WP:M, but it should really happen here. Thus, I cut-n-paste below my reply to a proposal to eliminate the section on classical mechanics.

Earlier versions of the article said, "the case of classical mechanics is handled in the subsection titled 'geometry', below." Two or three issues became apparent. One editor noted that "below" no longer makes sense with the new cell-phone navigation system. Another was that your edit removed all references to classical mechanics (flat tori are not classical mechanics, and also, they're trivial.) Third, there was some squirrely statement about how one cannot know how to move in a straight lines on a curved surface, or something equally weird. Looking at the edit history, I saw that it was you who added that remark. I concluded that you did not know Riemannian geometry, and so I used the word "geodesic" more than once, hoping it would set off a light-bulb. I also got the impression that you were unaware that classical mechanics is "just" symplectic geometry. So I tried to emphasize that, too. The motion of mechanical systems, as studied in classical mechanics, is given by solutions to the Hamilton-Jacobi equations. This is standard undergraduate college physics. So, here's the kicker: geodesics on Riemannian manifolds are given by solutions to the Hamilton-Jacobi equations on the tangent bundle. In this sense, motion on Reimannian manifolds is a special case of classical mechanics. This is because the tangent bundle is always a symplectic manifold.

Proving ergodicity is hard, so one always looks for simpler cases. The first case where ergodicity was proved in the non-trivial case is the Bolza surface, I think in the 1930's, which kind of launched the whole project of ergodicity in geometry. The next interesting results on "flat space" were Yakov Sinai's work in the 1960's(??) on a model, intended to approximate the atoms of a physical gas with hard elastic spheres. (Gasses are one of the classical topics in physics, and are used to illustrate all the basic thermodynamic relationships. Thus, being able to rigorously prove that a gas actually is ergodic is a big deal.) Sinai's system is now known as "Sinai's billiards" or (rarely?) the "hard-sphere gas". Many(?) other gases have been studied. If I recall correctly, the hexagon gas is exactly solvable (where the things bouncing around are hexagons. Something like that. (The hexagon gas might be a special case of the translation surface??) One can get the various thermodynamic parameters for it.) To summarize: geometry is a special case of classical mechanics. But you've cut all that out. 67.198.37.16 (talk) 15:06, 22 April 2023 (UTC)

Less important, but still worth reviewing, is that there are different types of ergodicity. Perhaps this should go in a section called "types of ergodicity". There are classification theorems that show that most "commonplace" ergodic systems are isomorphic to one of the Bernoulli schemes. There are countably many Bernoulli schemes. There are other classification theorems that show that, for certain kinds of systems, there are uncountably many different kinds of ergodicity. These are popularly called "anti-classification" theorems. "Anti-classification" is some attempt at humor: the systems are still classifiable; there are just uncountably many distinct classifications. (The ergodicity class would be "enumerated" by the infinitely long sequence of digits that specify a specific point on the Cantor set.) 67.198.37.16 (talk) 15:59, 22 April 2023 (UTC)


 * You ostensibly justify your efforts by wanting to make the page more accessible and yet you dismiss the flat torus as "trivial". So-called "trivial" examples are usually the most suitable to illustrate delicate concepts, as when well-chosen they allow to grasp the important part of the theory at once, and the flat torus gives a good idea about ergodicity of geodesic flows even if it is much less rich than the general case.
 * I'm sufficiently familiar with Riemannian geometry and the geodesic flow not to need to re-read your little ditty about it above. My point, that you did not address, was that these things have no immediate bearing on the topic of ergodicity of flows, which is the one discussed in this section.
 * I know what a geodesic is, and that the geodesic flow is defined on the unit tangent bundle rather than the manifold. Informally, this means that there are no globally defined directions on a Riemannian manifold, contrary to flat ones where one can study the geodesic flow independently in each direction; this was the intended meaning behind the remark. I don't think it is possible to convey these subtleties in detail in the short space it should occupy in the introduction of an article where flows are not the main topic.
 * Billiards and their relation to physics are certainly an interesting topic, but not one that should take too much space in this article ; hence i reduced them to a mere mention of flat surfaces. More detailed examples should not be given until later sections of the article (those which require a certain familiarity with the technicalities of the topic at hand to be correctly edited).
 * Your last paragraph above is particularly unclear and seems to refer to topics that do not belong in this article beyond short mentions. jraimbau (talk) 16:59, 22 April 2023 (UTC)


 * I attempted to structure the examples section so that it flows along natural lines. Here's my thinking. Most students in engineering, chemistry or physics are likely to first hear about ergodicity when studying thermodynamics. Since the earlier sections defining it in terms of measure theory are relatively abstract, it seemed important to connect the dots, to explain that the many-body view, and the more formal description is "the same thing".


 * Next comes classical mechanics, which is the foundational cornerstone of ergodicity, and where most of the precise mathematical results can be found. (i.e. Poincare and the small-denominator problem.) Here, I was thinking of the intended audience as being those who are thinking that "this has something to do with chaos and fractals", and have perhaps the double pendulum in the back for their mind, or perhaps have read about the Fermi–Pasta–Ulam–Tsingou problem. So, yes, here, perhaps I shifted gears to fast or too erratically. The flow that I like is the one that says "classical mechanics is just symplectic geometry; proving ergodicity is hard, here's a list of simpler, solvable variants." Since the section title is "classical mechanics", the toy problems should all be of that type. Nothing wrong with the flat torus, but its not classical mechanics, and so belongs elsewhere. Next, there is a natural segue "gee, all of these simplified classical mechanics problems are actually geometric, so lets take a look at what mathematicians have been able to achieve with flows on manifolds.".  It's a good place to re-emphasize from physics to math, since most of the results are formal, mathematical, and were obtained by mathematical physicists or "pure" mathematicians.


 * After that, more math, less geometric results, e.g. the Green-Tao theorem on arithmetic series. I cannot write that, because I have not followed the work there. FWIW, the Green-Tao theorem is an ergodic theorem, although that article doesn't actually say that. It would take some work to explain why. Ugh. I'd also love to quote work from  Jeffrey Lagarias and David Ruelle and whatever one might find in the journal of ergodic theory and dynamical systems. But I can't write that.  After that, the sections on quantum, economics, social sciences, although I am less thrilled about the last two; they should be moved to the article on the ergodic hypothesis.


 * Anyway, that's the proposal for the overall flow. We are arguing about how to transition from the more informal physics foundation, to the present-day mathematical physics, or even pure mathematics conceptions. Is the above outline suitable? 67.198.37.16 (talk) 23:07, 23 April 2023 (UTC)


 * (Footnote: the article on the ergodic hypothesis is terrible. There's almost no one at all working on wikipedia who is interested in dynamical systems. I get the impression that I created & wrote almost all WP articles on such topics. That's a pathetic statement about the involvement of experts in wikipedia. The fact that I also wrote much of the content on fiber bundles and spin is also alarming. There are lots of pop-sci enthusiasts covering basic topics in WP, and lots of grad students covering arcane topics in loving detail (Green-Tao theorem being a good example) but almost no one writing intermediate seminar-level articles. So, ergodic hypothesis is something a collage junior might get in a seminar; this article, on ergodicity, might be at the senior level or first-year grad-school level. But almost no one on WP is writing at this intermediate level. It's a prominent hole in article coverage. The loud thud of absolute silence you heard when you complained about me at WP:M is symptomatic of the threadbare level of involvement of experts. If it were not so, you wouldn't have to argue with me. 67.198.37.16 (talk) 23:56, 23 April 2023 (UTC))
 * I see better where you're coming from, though i still strongly disagree. For the record i think this article should mostly be about mathematics with a motivational part from physics and the article on the ergodic hypothesis should be only about the physics. I'll try to make a proposal including more physics, at the same time at some point i want to make more cuts to the too-long introductory part: this article should be about the mathematical concept of ergodicity and its applications to/appearances in the sciences, there is no need to give long developments about the context for the latter---there are other articles for this. jraimbau (talk) 18:40, 24 April 2023 (UTC)
 * Well, except maybe almost all research into ergodicity is done by physicists, and not mathematicians. I recall one math colloquium -- a presentation to the entire math dept., profs and grad students alike, that began with a sentence "There's this boring obscure corner of math called Hamiltonian dynamics, but if you dig into it, its not boring at all." and proceeded to review work by Anatole Katok and wife Svetlana (among other things). What I found surprising was how much the speaker felt she needed to review basic physics, stuff that undergrads would know as a matter of course. Compare to other colloquia speakers who promptly zoom away on advanced topics in (for example) algebraic topology with the assumption that the audience are already well-aware of the basics.
 * If you treat this as a mathematical topic, you will immediately loose 95% of your audience, who, I believe, are going to be mostly engineers, chemists, physics students, and the occasional financial guru. These are people who won't know what measure theory is, or what Borel sets are. So trying to turn this article into "pure math" is a mistake, I think. 67.198.37.16 (talk) 22:09, 24 April 2023 (UTC)
 * This is wrong, ergodicity is an important tool in many branches of contemporary mathematics. That anecdote you tell has nothing to do with this topic---Hamiltonian mechanics is but one of many fields where the concept has an impact, and the Katoks are definitely mathematicians and not physicists.
 * Ergodicity is a mathematical concept and the meat of the article should be about that. For those "95% of readers" there should be an introduction explaining the relevance of the concept to various sceinces, but as it stands this part of the article is way over-developed, the content should be limited to what's really relevant and the precise descriptions of each field should be left behind hyperlinks. If a scientist or engineer goes to this page they would immediately see whether it is relevant to their interests, and i'm sorry to say so but if you want to actually work with the concept you are going to need some familiarity with advanced mathematics.
 * The short first paragraph of the "finance" section does a good job of that---i can clearly see to which topics ergodicity is relevant, and should i want to know more i would just click on the links (and assuming the target pages are written comprehensively i would find more precise information about the role of ergodicity there). The next two paragraphs are a bad example---they should be way more quicker to explain why ergodicity might actually not be relevant at all to finance, instead they go on all sorts of tangents and end up hardly intelligible. In my view the sections on physics are too much like the latter. jraimbau (talk) 14:46, 30 April 2023 (UTC)

Split article into two?
Continuing from prior section: How about this as a solution: copy or move the bottom half of this article to a new article ergodicity (mathematics) that would then be free to accumulate the formal definitions of ergodicity? What would remain is that this article could be the sketchy, informal introduction, while the new article would contain not only formal definitions, but would be of the right format to grow over time with formal results and theorems. 67.198.37.16 (talk) 15:10, 29 April 2023 (UTC)


 * I completely disagree with this. If the article on ergodicity doe not clearly state what the modern definition of ergodicity is, and how to work with it, it can hardly be called encyclopedic. I will instead make the proposal that the physics contents should be pruned, and possibly moved to the article on ergodic hypothesis which as you noted above in is a very sorry state (as even a non-physicist such as i can see). jraimbau (talk) 14:50, 30 April 2023 (UTC)


 * OK, lets do this in phases. I will move 'ergodicity on finance' and 'ergodicity in social science' to the ergodic hypothesis article immediately, since both are obviously about assuming the ergodic hypothesis and not about ergodicity itself.


 * After this, we have sections titled:


 * Informal explanation
 * History and etymology
 * Ergodicity in physics and geometry


 * If I understand you correctly, you want all of these moved somewhere else, and not here. How about a new article ergodicity (dynamical systems)? These sections don't belong in ergodic hypothesis, since none of them have anything to do with the hypothesis. The hypothesis merely states that "lets assume that actual physical systems are ergodic". This is distinct from proving that, say, the flow on some elliptic curve is ergodic, and that it's spectrum is some Bessel function, etc.; these statements are proofs, not hypothesis, and thus wouldn't go into the ergodic hypothesis article. 67.198.37.16 (talk) 19:12, 30 April 2023 (UTC)