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In his classic text, Callen gives the key postulate (Postulate II) as

There exists a function (called the entropy $S$) of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of an internal constraint are those that maximize the entropy over the manifold of constrained equilibrium states.

My question is, should this postulate be tacitly understood to apply only to closed (isolated) systems?

Tobias Fünke
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EE18
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  • Since it's an extensive function we can always scale the size of the system without any change of the definition, can't we? So now it depends on whether we define such a scaling as being within the definition of "closed (isolated) system". Maybe I am missing something. – FlatterMann May 26 '23 at 19:45
  • I'm not sure I follow what that has to do with the question? @FlatterMann – EE18 May 26 '23 at 19:50
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    As stated, the maximum of $S$ is understood only for isolated system. And I am actually surprised that Callen did not state thus explicitly. Also, isolated in regular usage is not the same as "closed". I think you will find these comments also helpful https://physics.stackexchange.com/q/534173/ and https://physics.stackexchange.com/q/761468/ and https://physics.stackexchange.com/q/762504/ – hyportnex May 26 '23 at 21:56
  • I am simply saying that even if we assume that the system is isolated and closed, the extensive parameters scale proportionally to the system size, so definitions like these, which only require some form of an extremal principle do not rule out scaling. At least that is my understanding. The only way to rule out rescaling would be to say that the system has to be extremal and limited to a unique value of S, which might run into other problems (which I didn't think through). – FlatterMann May 26 '23 at 22:26
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    I think what @FlatterMann is alluding to is that the entropy maximum principle as stated in Postulate II by itself is not enough, it has only additivity in it for one system separated in to various parts by the constraints. Scaling is an independent property that will be needed to show the convexity of the entropy $\cap$ and of the potentials $\mathbf {\cup}$. This issue is in a later Chapter 8 of Callen. – hyportnex May 26 '23 at 22:56
  • @hyportnex It's good to know that Callen discusses the subtle consequences explicitly. I really need to read his work. – FlatterMann May 26 '23 at 23:03
  • Thanks to both of you for your answers. I will have to think about it a bit more because I'm still not sure I follow what scaling has to do with the question. If you're able to flesh it out as an answer I would appreciate it. @FlatterMann – EE18 May 27 '23 at 01:33
  • and @hyportnex. I really appreciate the help as always! – EE18 May 27 '23 at 01:33
  • just read the first 5 pages of Chapter 6 and the article by Leib&Yngvason https://www.ams.org/notices/199805/lieb.pdf – hyportnex May 27 '23 at 02:15
  • Related (Callen II postulate) Entropy maximum postulate and reversibility. – Quillo May 27 '23 at 12:56
  • @Quillo The question you linked is very interesting and unanswered. Would you happen to know? – EE18 May 27 '23 at 16:48

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You are right. Callen's postulate II applies to closed systems. Even though Callen does not explicitly mention this requirement in its postulate, it should be clear by reading previous pages.

Indeed, Postulate II is in section 1.10 (I am referring to the second edition of his book). The starting paragraph clearly states that the postulates provide the simplest formal solution to the basic problem of thermodynamics. The basic problem explicitly defined in the previous section is the determination of the equilibrium state that eventually results after the removal of internal constraints in a closed, composite system (bold type is mine).

GiorgioP-DoomsdayClockIsAt-90
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