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In the derivation of extensivity of entropy for the micro-canonical ensemble, we assume an ensemble of two systems with the energies $E_1$ and $E_2$. The total energy is given as $E<E_1+E_2<E+\Delta$.

Further on in the derivation, in order to approximate $\log{\Gamma}$ and $\log{\frac{E}{\Delta}}$, we use the relations

$\underline{\log{\Gamma}\propto N}$ and $\underline{\log{\frac{E}{\Delta}}\propto \log{N}}$

with the phase space volume $\Gamma=\int_{E<H(p,q)<E+\Delta} \frac{1}{N!h^{3N}}\mathrm{d}p^{3N}\mathrm{d}q^{3N}$ Textbooks like Huang's Statistical Mechanics do not motivate or derive these relations.

With only the knowledge of $\Gamma(E)$ given above, get to the relations $\log{\Gamma}\propto N$ and $\log{\frac{E}{\Delta}}\propto \log{N}$?

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  • $\begingroup$ Could you explain exactly what you mean by each of your variables? You have an $E$, an $E_1$, an $E_{1i}$, etc and it is hard to infer what each of these is meant to be. $\endgroup$ – By Symmetry Oct 13 '18 at 8:49
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1) I assume you are following a derivation in Hunag's textbook, so I will try to stick to it. I don't think his phase space integral is normalized by $\frac{1}{N!h^{3N}}$ (eq. 6.10 in 2nd edition).

It is an 6N-dimensional volume. So no matter what is the exact shape of this 6N-dimensional volume, it scales as something to power N. Log of this value scales as N.

Alternatively, you can assume integrating over a 6N dimensional sphere, and the volume of it is $R^{6N}$.

2) Last approximation requires to say that energy $E$ scales as $N$ and $\Delta$ is a number that does not depend on the number of particles.

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