Extensiveness of entropy in classical microcanonical ensemble

Question

In introducing microcanonical ensemble of classical statistical mechanics one pretty much starts by postulating that entropy of the system has the form

$S(V,E) = k \log \Gamma(V,E)$,

where $\Gamma$ is appropariate measure of available volume in phase space, with a few possible choices. I am not spelling out the details because I assume these are known, but I remark that I use the definitions given in the classical book "Statistical mechanics" by K. Huang. To justify identification of this quantity with thermodynamical entropy one, among other things, shows that it is extensive. Now again this proof is probably well known and can be found in any book in stat. mech. so I don't quote it. The problem is as follows: after introducing all this Huang uses this definition to calculate entropy of ideal gas. He discusses that there is some problem in that this expression cannot be true due to the Gibbs paradox and then introduces the so called correct Boltzmann counting. Now physical origin of this (as classical limit of bosonic/fermionic statistics) is clear. However what bothers me is that entropy obtained before introducing the correct Boltzmann counting is not extensive (it is after the correction). I remind that necessity of Boltzmann counting was not taken into account in the proof of extensiveness of entropy. Therefore it seems that there was some mistake in the proof. It seems that this is a counterexample! This bothers me because it is a foundational concept which seems to break down in the simplest possible case. Is there some problem in my reasoning?

Answer 1

The result provided in the book by K.Huang is only a partial answer to the request of extensiveness of the entropy. The limit of his derivation is that, in the combined system he analyzes, only energy can be exchanged while the number of particles in the two sub-volumes ($N_1$ and $N_2$) are kept fixed. It would correspond to a physical situation where a wall impermeable to fluxes of particles would remain after combining the two separate systems. This is not the most general case for discussing the thermodynamic extensiveness. The reason the final result which is puzzling you is that it is not consistent with the right behavior of the entropy with respect to variation of the number of particles. But this is a well known problem of the formula $S(V,E) = k \log \Gamma(V,E)$ which should be written more correctly as $$S(V,E,N) = k \log \Gamma(V,E,N) + \phi(N)$$ where $\phi(N)$ is a function of the variable $N$ only (see R.Swendsen, Entropy,10, 15 (2008)). One way of determining the value of such function is precisely a careful analysis of the conditions for the extensiveness. A complete treatment of the extensive property of the microcanonical entropy can be found for instance, in the book "Statistical Mechanics and Applications in condensed Matter" by C.Di Castro and R. Raimondi. The analysis of the different ways of dividing $N_1$ and $N_2$ particles in the two sub-volumes introduces in a natural way the $1/N!$ factor required to avoid the Gibbs paradox. Basically, one has to write the microcanonical entropy of the combined system as $$ \sum_{N_1=0}^N \frac{N!}{N_1!(N-N_1)!} \sum_{E1} \Gamma(V,E_1,N_1)\Gamma(V,E-E_1,N-N_1) $$ where the binomial coefficient counts the number of ways $N_1$ particles can be assigned to system 1 and ($N-N_1$) to system 2. So, from a more careful analysis of the general condition for the validity of the extensive property one can avoid the apparent flaw of Huang's argument you correctly pointed out.