Understanding Kerson Huang's approach to the extensive property of the entropy in statistical mechanics I am a Mathematics student preparing myself for the exam in biological models. A part of the lectures was about statistical mechanics. After a brief introduction (hamiltonian, microcanonical ensamble) we proceeded to the definition of the entropy and its properties. There was some part I did not understand then and reading a textbook (Kerson Huang - "Statistical Mechanics", 6.2, p. 133) I am still confused. 
Why should we take the values $$\bar E_1, \bar E_2 $$ that maximise the function $$ \Gamma(E) = \Gamma(\bar E_1)\Gamma(\bar E_2) $$
and why does the following $$ S(E,V) = S(\bar E_1,V_1)+S(\bar E_2,V_2)+O(\log N)$$
prove the extensive property of the entropy? 
For me it proves only the special case with energies that maximise the function. 
I saw a similiar approach in prof. David Tong's lecture notes on his website, but I was not able to understand them. I think I am missing some fundamental assumption or do not understand the gamma function properly. I do not have strong background in physics, so would appreciate a detailed answer.
 A: Before being  able to exchange energy the two systems will have definite energies $E_1$ and $E_2$, and so definite entropies $S_1(E_1)$ and $S_2(E_2)$. This derivation doesn't show that the entropy of the final 1+2 system is $\bf S_1(E_1)+S_2(E_2)$. In fact, it can't show this is the case because it's not true (why?).
What the derivation does prove is that


*

*when two different systems are allowed to exchange energy then at equilibrium the two systems will reach definite energies $\bar{E_1}$ and $\bar{E_2}$, that may be different from the initial energies (but satisfy the condition $E_1+E_2=\bar{E_1}+\bar{E_2}$)

*the entropy of the system 1+2 at equilibrium is precisely equal to the sum of the entropies of the two single systems, when they have the equilibrium energies $\bar{E_1}$ and $\bar{E_2}$

*this entropy is the maximum possibile entropy that the 1+2 system can have
What is meant by the extensiveness of entropy is what is addressed by point number 2.
A: $\bar E_1, \bar E_2 $ are the most probable values. You can think about that in different ways, they are the most probable values because they maximize $ \Gamma(E) = \Gamma(\bar E_1)\Gamma(\bar E_2) $ and therefore they maximize the entropy, according to the Principle of Maximum Entropy
If you understand why maximizing $\Gamma$ you maximize the entropy you can skip the next paragraph(2 questions with answers), but Since you said you don't understand the $\Gamma$ function I'll try to add some notions and relate them to the question.

*

*So what is $\Gamma(E)$?
It is the number of microstates available to the system, or if you want a more formal definition, it is the phase space volume occupied by all the reference points of the system. So it is basically a volume in phase space which represents how many microstates system can be in. Note that all these microstates correspond to the same physical macrostate.


*How does that relate to the question?
It does considering that the entropy of the system is the $\ln \Gamma(E)$, or more precisely, it is the logarithm of $\frac {\Gamma}{Planck \, constant }$.  In this way you obtain a pure number inside the logarithm.  In classical statistical mechanics you can't prove that you have to divide for the Planck constant. In this framework you just divide for a constant of the dimension of an action (phase space volume), the constant is then fixed by quantum statistical mechanics.
That said, I repeat that those values of the energy  $\bar E_1, \bar E_2 $ are the most probable values, so Kerson & Huang's derivation proves the extensive property of entropy for all the systems that respects the principle of maximum entropy, that is as far as i know, every known system.
P.S.
I don't remember the details of their proof, but I remember that to be carried out, that proof needed an important assumption, which if I'm correct is implicit in their textbook, that is for a typical system, the number of microstates $\Gamma(E)$ pertaining to any macrostate that departs even slightly from
the most probable one is “orders of magnitude” smaller than the number pertaining to the latter, which is $\Gamma(\bar E)$. Without this you couldn't do some mathematical passages in the proof.$
