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For personal reasons, I had to leaf through various documents relating to statistical mechanics. It's not at all my specialty and I have trouble understanding certain notions; for example, I found the following text (which I imaged), from the book Topics in Statistical Mechanics [Brian Cowan] (2005 edition):

enter image description here

I understand the difference between microstate and macrostate, but I don't see what the author means in this passage. If the isolated system he is talking about has a fixed energy E, volume V and number of particles N, then there can only be one macrostate (E,V,N), right? That is, all microstates in this system should lead to the same macrostate, right? So why does he speak of "given macrostate" as if there could be several, made by different configurations of microstates?

This is one of the questions that are not absolutely essential to solve in order to progress in the theory, but I find this difficulty of understanding a little embarrassing... Could someone tell me if the expression of the author implies things that I would not have understood, if it is a kind of shortcut that I do not know how to interpret, if what he says is actually perfectly clear?

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    $\begingroup$ Please do not post images of texts you want to quote, but type it out instead so it is readable for all users and so that it can be indexed by search engines. For formulae, use MathJax instead. $\endgroup$
    – hft
    Dec 15, 2022 at 21:52
  • $\begingroup$ You can use the ">" character at the start of a line to create a quote. $\endgroup$
    – hft
    Dec 15, 2022 at 21:54
  • $\begingroup$ The author may be gearing up to explain the canonical ensemble. The ensemble with E, V, N fixed is called "microcanonical." The ensemble with T, V, and N fixed is called "canonical." When deriving the state probabilities for the canonical ensemble, one usually starts by looking at two coupled systems (one called a "heat bath") that together are microcanonical, but the other system (the one that is not the heat bath) is the system of interest and it can exchange energy with the heat bath. $\endgroup$
    – hft
    Dec 15, 2022 at 22:02
  • $\begingroup$ Isn't he comparing different macrostates, e.g $( E_1, V_1, N_1)$ with $( E_2, V_2, N_2)$ ? They will no doubt have different numbers of microstates and therefore different probabilities. $\endgroup$ Dec 15, 2022 at 22:35

3 Answers 3

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I understand your confusion. Given an isolated system, the variables $(E,V,N)$ will always stay constant if we let it evolve in time. When the author says for a "given macrostate" he is referring to a more general system that can be parametrized by $E,V$ or $N$. For example, my 'general system' could be an ideal gas. This theoretical gas could have any number of particles and have any total energy. Once I specify all these three variables I will have chosen one particular system from my 'general system', i.e. from the space of all possible ideal gasses. This gas will then have the same values $(E,V,N)$ for all eternity.

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It is true than an isolated system will only have one macrostate accessible. However, the sentence in the text beginning with, “Then from the Fundamental Postulate…” is not referring to a situation where there is an isolated system.

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  • $\begingroup$ Thank you for your remarks. I have no competence to criticize the author, but I deduce that his expression can lead to confusion: he seems to speak at the same time of the particular case of a system where everything is fixed and where any configuration of microstates leads to 1 macrostate (E,V,N) determined, and the well-known general case where the probability of a macrostate is a function of the number of microstates which realize it. I agree with the interpretation of AccidentalTaylorExpansion. $\endgroup$
    – Andrew
    Dec 16, 2022 at 8:56
  • $\begingroup$ Now the confusion seems to be cleared up if we follow the remark of Jbag1212. My conclusion is that the phrase “Note, […] (extensive quantities)” is simply not in the right place… The rest is clear. Would you agree with that? $\endgroup$
    – Andrew
    Dec 16, 2022 at 8:56
  • $\begingroup$ Yes, I agree there is the potential for confusion. I think the best way to read what the author wrote is to imagine the sentence, “Note, in particular, …” as being surrounded by parantheses. $\endgroup$
    – Jbag1212
    Dec 16, 2022 at 18:46
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Imagine a box of N number of atoms. The macroscopic states can be parametrized by the density of atoms as a function of spacial points. The overall energy, volume and atoms number are the same, but they can still have different particle number distributions.

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