Talk:Statistical ensemble (mathematical physics)

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Having taught statistics at four universities, including one renowned throughout the world, I have not encountered the phrase "statistical ensemble" used in this sense. Certainly populations and samples are mentioned all the time, but when is this term ever used except by physicists? Michael Hardy 22:27 Feb 17, 2003 (UTC)


Yep, you're right, it's a physics term. Let's edit.

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[edit] Statistical ensembles

Here's a suggestion to make the statistical ensemble article more consistent with the quantum statistical mechanics article. Two relevant operational notions on ensembles are

  • Testing statistical equivalence of ensembles
  • Probabilistic sampling from ensembles A, B with probability p 1- p to form the convex combination of ensembles.

Since in the quantum mechanical case, ensembles are modeled by density operators, this is a cleaner approach to the pure-state mixed-state dichotomy, because equivalence classes of ensembles have the structure of a convex set.

Moreover I like to think of the concept of ensemble as being itself operationally defined in the following sense: there is some laboratory procedure for producing systems which are instances of this ensemble. By repeating this laboratory procedure we obtain a sequence of systems X1, X2 .... Xk... Exactly how this sequential ensemble defines a state might be left out of the article, but here is an explanation: A sequential ensemble defines a state by the time average of measurements: e.g. for each quantum yes-no question E (for reference this is discussed in the article quantum logic) we obtain a sequence of measured values (e.g. by observing a dial on a gauge) Meas(E, Xn). Each one of these values is a 0 or 1. (Of course the measurement process alters the system so it no longer is an instance of the ensemble)

Assume the time average exists

\sigma(E) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N \operatorname{Meas}(E, X_k)

This is a kind of ergodicity assumption, but for explaining the operational meaning of ensemble I think this is good enough. Using Gleason's theorem this is given by a density operator S:

\sigma(E) = \operatorname{Tr}(E S)

[edit] Climateprediction & Grand Ensemble

I'd like to see extra added to this page to include mention of climate prediction.

See Ensemble in BOINC Wiki

Also how about defining a grand ensemble as an ensemble of ensembles. Perhaps include a diagram such as at a)

http://www.opendemocracy.net/content/articles/2571/images/Fig_1.gif

from http://www.opendemocracy.net/globalization-climate_change_debate/2571.jsp

In fact there is a lot of complex stuff in this article which I think would be better placed in a referenced page for Using Statistical Ensembles. Keep this page just for saying what an ensemble is. that is just MHO.

[edit] New article

The new changes to the article regard a very unusual interpretation of statistical ensemble. I am reverting. If there is indeed a usage in meteorology which is different, then this should be place in another article.--CSTAR 15:45, 11 Jun 2005 (UTC)

[edit] Most moved to Using Statistical Ensembles

I have done this suggested move now.

Sources for new material include http://www.climateprediction.net/science/strategy_adv.php http://www.opendemocracy.net/globalization-climate_change_debate/2571.jsp http://www.climateprediction.net/science/pubs/nature_first_results.pdf.


[edit] mce, ce & gce

Whatever else, it should be possible to access the usual classical statistical ensembles used by physicists and chemists, namely:- microcanonical ensemble, canonical ensemble, grand canonical ensemble. There should be short definitions of the three here, too. This appears to have got lost (again?) Linuxlad 15:56, 11 Jun 2005 (UTC)

[edit] Idealization and "mental" copies

Could we use a different expression other than "mental". Isn't it enough to say it is an idealization? --CSTAR 13:59, 24 October 2005 (UTC)

The word idealization on its own is not clear. For example, it would be an idealization to consider the system to be made up a large number of real subsystems, each statistically independent. This is the idealization used by Boltzmann in 1872 to derive the Boltzmann equation, the H-theorem and the Maxwell-Boltzmann distribution. But in 1878 Gibbs goes beyond this by considering the probability distribution for the system as a whole. There is now no longer a population of real subsystems: there is now only one real system, in one (unknown) state. But because it is useful to imagine probabilities in terms of drawing an individual from a population, Gibbs introduces a mental population of possibilities, which he calls an ensemble.
(Note also that something like the Boltzmann distribution is exact when we can consider it as a probability, ie the result of deriving it for an infinitely large mental ensemble; but only approximately correct at any one time, and subject to fluctuation, when we consider it as a fraction of a real population).
IMO it is thus important that the word "mental" (or something very similar) should remain in the article to flag this point. Jheald 17:57, 27 October 2005 (UTC)
OK, you make a valid point, but I still prefer a better phrasing. I don't like the association with "minds"; how about "imaginary"? or "imagined". If you do not like these, please give this some thought and try to find another expresion. I will also think about better formulations.--CSTAR 18:01, 27 October 2005 (UTC)
"Imaginary" I don't like, because of the association with complex numbers. "Imagined" I could live with; but I think "mental" is better, because (IMO) it's simpler and more immediate. Is there a problem in the association with "mind"? -- That is where the ensemble lives. Jheald 20:37, 27 October 2005 (UTC)
I think there is a problem associating it with minds, because that suggests that if minds didn't exist then neither would ensembles. I'm not trying to be picky or argue for argument's sake; I would prefer a formulation which is consistent with other similar abstractions in statistics or mathematics. Sample space of a stochastic process, such as the Wiener process is a similar abstraction. Choosing terms carefully will have a payoff, I think. But I concede your larger point.--CSTAR 03:08, 28 October 2005 (UTC)
Would "similar" or "diversified" suffice? I don't think there is a requirement for the copies to be exact, just proprogated. GChriss 21:02, 8 February 2006 (UTC)
No. I think you're thinking about the practical running of an ensemble of climate models. But that's really only a derived use of the word -- fundamentally it's called an ensemble because it's supposed to be a collection of states (and dynamics) that we've dreamed up which we hope may be representative of the state the world might be in.
The point (at least in statistical mechanics) is that the copies are not real. They do not exist in the real world. They are a thought-experiment, as one considers all the different states the system might be in, which defines a set you can then do analytical calculations on. -- Jheald 00:09, 9 February 2006 (UTC).
Right. But I still don't like "minds" in this definition :)-CSTAR 00:12, 9 February 2006 (UTC)

[edit] comment on article.

i've made some edits and supplied a few details. the intro of the article is still rather sloppy. Since there's "mathematical" in the title, that should be the context of the article. not some stuff on, say, meterology. surely that could be mentioned as a side remark, but that should not interrupt the coherence and continuity of the mathematical presentation of the article, and that's what the non-mathematical remarks in the present version are doing. Mct mht 07:54, 24 May 2006 (UTC)