Derivation of Proportionality of Phase Space Volume log(Γ)∝N

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.

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 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 $$\log{\frac{E}{\Delta}}\propto \log{N}$$?

• 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. Oct 13 '18 at 8:49

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).
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.