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In my lecture notes, the Boltzmann entropy form was motivated as a valid form for entropy partly because it was extensive. However this hinges on the assumption that the statistical weight for a macrostate of two systems combined is their product. Why is this valid- for large systems at least- and when might it break down?

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I think that the explanation here is the same as in classical thermodynamics: the interaction between the two subsystems is a "surface effect".

Suppose the two subsystems are cubic, of volume $L^3$, sharing an interface of area $\sim L^2$. If the interactions are of short range, $d$, the perturbation on either system (just from counting interactions) has a relative magnitude $\sim dL^2/L^3=d/L$. Of course, this has an effect on the number of states $\Omega(E)$ in both subsystems having energy $E_1$ or $E_2$ respectively, but bearing in mind that the logarithm of $\Omega$ is the important quantity, this will be small provided $d\ll L$. So, one can make the approximation that the systems are statistically independent.

One can expect the argument to break down if the interactions are long ranged (when extensivity of thermodynamic properties does actually come into question) and when the system is small (when one would not expect it to hold). Also, if the subsystems are chosen to have an unusually large interaction area: for example, the oppositely coloured lattice sites in a checkerboard colouring of Ising-like models.

There are various other subtleties associated with the extensivity of the Boltzmann entropy, some of which can be found discussed on Physics SE, but I think this is the one you were worried about.

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  • $\begingroup$ Thank you for your reply! And I see the slight subtlety here. I took the extensivity of entropy as true, but was troubled by the multiplicity of statistical weights (which would be iff entropy is extensive in this case, and the entropy does indeed depend on the statistical weight in this way). In reality, the extensivity of the entropy is debatable, if I understand correctly. Please correct me if I am worng $\endgroup$ – Meep Nov 29 '18 at 12:35
  • $\begingroup$ And I suppose the other subtleties include mixing systems as well, as opposed to simply letting their surfaces come into contact. $\endgroup$ – Meep Nov 29 '18 at 12:36
  • $\begingroup$ There are systems with long-range forces for which the extensivity of thermodynamic properties in general comes into question. There is a link to a paper on this in my answer physics.stackexchange.com/a/434303/197851 to another question. However, I want to stress that the more usual situation is that the entropy and other thermodynamic quantities are extensive, because most interactions are short-ranged (as defined in that paper). My answer is rather "handwaving", but the subtleties of the thermodynamic limit are discussed more formally in papers and books (e.g. by David Ruelle). $\endgroup$ – user197851 Nov 29 '18 at 13:09

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