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I am not a physicist, and I'm mainly used to thinking about probability spaces (formulated in measure theory).

I am trying to understand what a "statistical ensemble" is, and how it is different from a probability space in terms of measure theory.

In measure theory, a probability space is simply a tuple $(A,\Sigma, P)$, where $\Sigma\subseteq \mathcal P (A)$, and $P:\Sigma\to [0,1]$. My guess would be that an ensemble is simply a subset $A^*\subseteq A$, where $A$ is interpreted as the state space of a physical system, and $A^*$ is a subset of that state space where we have that a certain function(s) has a fixed value(s). For example, take the total energy of the system $E:A\to [0,\infty]$. We might let $A^*=\{a\in A: E(a)=E^*\}$ for some constant $E^*$. Assuming for a moment that we are dealing with a quantum mechanical system so the state space is finite, we can then define a probability measure $P$ on the measurable space $(A^*, \mathcal P(A^*))$, such that for all $a,b\in A^*$, we have that $P(a)=P(b)$. (This is then the fundamental postulate of statistical mechanics, if I'm not mistaken). Thus in this case, our ensemble would be the probability space $(A^*, \mathcal P (A^*), P)$.

Questions:

  1. Is my understanding correct?

  2. Is there a more subtle difference between the two?

  3. Is there a good reason that "ensemble" is terminologically different from the terms used in probability theory, or is this purely a historical quirk? What would be the equivalent term as used by probability theorists?

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    $\begingroup$ Yes, from a practical point of view, statistical ensembles are just probability spaces, and the use of two different terminologies is mainly due to historical reasons. The notion of probability space was introduced in the 1930s by Kolmogorov (if I am not mistaken; there may have been precursors), while the notion of statistical ensemble was introduced by Boltzmann in the 1880s (ergode and holode) and, with the modern terminology, by Gibbs a bit later (and inspired by Boltzmann). $\endgroup$ – Yvan Velenik Feb 22 '18 at 7:54
  • $\begingroup$ @YvanVelenik I think this could be an answer :) $\endgroup$ – valerio Feb 22 '18 at 10:21
  • $\begingroup$ @Valerio: if no other answers appears and if I find time to improve on my above comment (in particular to check dates and provide sources), I may convert it to an answer in a few days. $\endgroup$ – Yvan Velenik Feb 22 '18 at 10:47
  • $\begingroup$ @YvanVelenik The only thing I would recommend is to check the statement about probability theory...Kolmogorov gave one of the first rigorous formalizations, but I am not sure if he was the first one to do so. $\endgroup$ – valerio Feb 22 '18 at 10:49
  • $\begingroup$ @YvanVelenik, thank you :). I am still somehow a bit skeptical though that there is not some subtle difference. If you were to turn your comment into an answer, I hope you can also address if there are any subtle differences (even if broadly speaking they are the same). $\endgroup$ – user56834 Feb 22 '18 at 13:39
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From a practical point of view, statistical ensembles are just probability spaces, and the use of two different terminologies is mainly due to historical reasons.

The (general, abstract) notion of probability space was introduced in the 1930s by Kolmogorov in his famous work. (Of course, particular instances of this notion had already been used before.)

The notion of statistical ensembles (which he called monodes, special cases being ergode (=microcanonical) and holode (=canonical)) was introduced earlier by Boltzmann in the 1880s. The modern terminology (microcanonical, canonical, grand canonical) was then introduced by Gibbs some years later (and was of course inspired by Boltzmann).

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