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On the entry of Gibb's entropy formula on Wikipedia, the following definition is given: "The macroscopic state of the system is defined by a distribution on the microstates that are accessible to a system in the course of its thermal fluctuations." (I will assume that this description is correct. If not, this question does not apply.)

I'm having a hard time grasping exactly what this description tries to say. I initially thought that the distribution of microstates was given by which microstates that could instantiate the macrostate in question. However, both "accessible" and "in the course of its thermal fluctuations" makes me wonder about if this interpretation is correct.

When reading the description above, it seems to me that you should start of with a certain microstate that instantiate the macrostate and then see how many states that are accessible, that is, that could be the case within the time course of some time unit defined by the thermal fluctuation...Ehh....I might have lost myself there.

Could somebody deconstruct the description cited above?

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3 Answers 3

The term "accessible" is used in a somewhat sloppy way in the article (used sometimes, sometimes omitted). I think it's to be understood implicitely that microstates are accessible, i.e. they can evolve within a connected set. The macroscopically constrained set is called a statistical ensemble, for example of constant energy (so an example of a non-accessible microstate would be a state with another energy and shouldn't be counted then). This is obvious if the microstates are generated incrementally from each other but not obvious if you're generating them by randomizing configurations (see the interesting concept of the ergodic hypothesis).

So your quote says that the macroscopic state is a weighted collection of microstates, and that they have to be physically realizable from your starting conditions. The stuff about "thermal fluctuations" means you can't have for example a magic force you can enable at a point in time which pulls all the atoms of a gas into the corner of the room - I guess another way of putting it is that the walk through the configuration space of the system has to be a random walk.

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The problem is spontaneous symmetry breaking, e.g. ferromagnets. In any finite system, there cannot be symmetry breaking, because there is a finite (but small) chance that you go from one macroscopic magnetisation to another. But in an infinite system (i.e. "thermodynamic limit") this probability falls to zero. In quantum mechanics language this is often called a "superselection sector", but is essentially the same idea. Thus you need to construct ensembles which respect this symmetry breaking. By requiring the ensemble to be "thermally accessible" is a natural way to do this.

To be clear, this can be seen as an intellectual exercise to allow commuting of various limits; specifically, to take the thermodynamic limit first, as opposed to last.

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The description as given in Wikipedia is correct only when the system is in full thermodynamic equilibrium (although the page explicitly claims validity in general), and only if you disregard the last part of the statement.

In equilibrium, the density matrix commutes with the energy, each energy eigenstate $E_i$ has a well-defined probability $p_i$ in the corresponding ensemble, ''accessible'' means being present with a nonzero probability, and the formula $S=-k_B \sum_i p_i\log p_i$ is valid.

But the talk about ''in the course of its thermal fluctuations'' is superficial talk without any formal substance. It suggests a (very questionable) background assumption that the system actually fluctuates in time by hopping between the eigenstates, for which the machinery of statistical mechnaics gives not the slightest support.

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If you are confident that you are right about wiki being wrong, you should edit that wiki article. –  Siyuan Ren Jun 19 '12 at 6:13
This is not that questionable classically--- the system does wander around in phase space classically. In any reasonable definition of the quantum state of the system, any infinitesimal interaction with the environment will lead it to have statistical densities on different states, and this is the analog of the classical wandering around, so I think the warnings are overblown, although I do think it is better to have a-priori probabilities. –  Ron Maimon Jul 18 '12 at 18:20
@KarsusRen: Correcting everything that is wrong in the WWW is an endless job, but not my mission in life. There are those dedicated to Wikipedia but I am not one of them. –  Arnold Neumaier Jul 18 '12 at 19:08

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