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Im studying statistical mechanics and came across the ensembles.

  1. Now system of large number of particles can be defined by an ensemble which contains elements (infinite of them) where each element is the mental copy of system at a particular time and time average of any quantity of system can be assumed as same as ensemble average (avg over these mental copies)
  2. In another book I saw that an ensemble can also be defined as a collection of a very large number of assemblies which are essentially independent of one another but which have been made macroscopically as identical as possible.

Now my doubt is that the element in the 1 case is same as assembly in case 2 or different? How they differ from one another?

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Ensemble is many systems, evolving from different initial conditions. Ergodicity, an important assumption in statistical physics reasoning, is that time averaging (for a single system) is equivalent to the ensemble averaging (over many systems.)

Note that the same definition of ensemble is used in quantum mechanics, where it cannot (generally) be replaced by averaging over a single system (since a measurement collapses wave function.)

Ensemble (mathematical physics)

In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a set of systems of particles used in statistical mechanics to describe a single system.

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    $\begingroup$ Importance of quasi-ergodicity is controversial. I suppose it is important in ergodic school of statistical mechanics. But it is not in the pragmatist/subjectivist school of Gibbs and Jaynes. $\endgroup$ Feb 20, 2023 at 23:47
  • $\begingroup$ @JánLalinský good point. As far as I understand, Jaynes' approach is Bayesian - he doesn't need an ensemble at all. In the frequentist framework one needs to justify that the ensemble averages are describing the single system at hand. $\endgroup$
    – Roger V.
    Feb 21, 2023 at 5:56
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They are just the same. This ensemble definition is a bit archaic and was useful back in the days in order to visualize the Ergodic principle ($\lim_{T\rightarrow \infty}\frac{1}{T}\int^{t=T}_{t=0} dt \ ...=\frac{1}{Z}\int_{\Gamma} \ d\alpha \ ...$ ).

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