Talk:Statistical ensemble

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.

Contents

1 Statistical ensembles 2 Climateprediction & Grand Ensemble 3 New article 4 Most moved to Using Statistical Ensembles 5 mce, ce & gce

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 .... X_k... 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, X_n). 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

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

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:

<math> \sigma(E) = \operatorname{Tr}(E S) <math>

Climateprediction & Grand Ensemble

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

See Ensemble in BOINC Wiki (http://boinc-doc.net/boinc-wiki/index.php?title=Ensemble)

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

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.

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)

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.

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)