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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)
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?
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)
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)
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)
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))
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)
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)
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)
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)
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" [1] 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)
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. [2] Neugierigxl ( talk) 14:40, 10 May 2015 (UTC)
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.
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)
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.
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:
0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 |
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)
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 |
is ergodic, but you cannot go from one state to any other state in "one" step. Arsik87 ( talk) 23:03, 17 June 2018 (UTC)
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))
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
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)
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)
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:
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)
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)
The definition of ergodicity in the Informal Explanation section is currently given as
If a set eventually comes to fill all of over a long period of time (that is, if approaches all of for large ), the system is said to be ergodic.
But this is a description of mixing, not ergodicity. does not need to approach , 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)
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)
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)
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)
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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)
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?
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)
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)
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)
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))
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)
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)
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)
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)
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" [1] 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)
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. [2] Neugierigxl ( talk) 14:40, 10 May 2015 (UTC)
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.
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)
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.
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:
0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 |
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)
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 |
is ergodic, but you cannot go from one state to any other state in "one" step. Arsik87 ( talk) 23:03, 17 June 2018 (UTC)
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))
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
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)
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)
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:
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)
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)
The definition of ergodicity in the Informal Explanation section is currently given as
If a set eventually comes to fill all of over a long period of time (that is, if approaches all of for large ), the system is said to be ergodic.
But this is a description of mixing, not ergodicity. does not need to approach , 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)
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)
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)
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)