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Implications of ergodicity[edit]

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. (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?

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.[1] Neugierigxl (talk) 14:40, 10 May 2015 (UTC)

Expert help: intuitive definition[edit]

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)
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. (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 ( (talk) 00:55, 19 March 2015 (UTC))

Requested move[edit]

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[edit]

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":

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 (talk) 22:59, 31 January 2013 (UTC)

External Source Improvement[edit]

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 (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 electronicsGroovamos (talk) 10:16, 10 March 2014 (UTC)[edit]

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.

  1. ^ Oxford Dictionary of Statistics, ISBN 978-0-19-954145-4