# Why does Gibbs' factor appear in the partition function of Maxwell-Boltzmann statistic?

In the assumption of Maxwell-Boltzmann statistics, particles are treated as distinguishable. So for identical particles with discrete energy levels, there are degeneracies for any specific occupation state: $$|n_1,n_2,...\rangle$$

$$\frac{N!}{n_1!n_2!n_3!...}$$

The partition function in the Maxwell-Boltzmann statistics is:

$$\begin{split} Z=tr(\exp{-(\beta H)})&=\frac{1}{N!}\sum^{N}_{n_1,n_2,...=0}'\frac{N!}{n_1!n_2!n_3!...}\langle n_1,n_2,...|\exp{-(\beta H)}|n_1,n_2,...\rangle\\ &=\frac{1}{N!}\sum^{N}_{n_1,n_2,...=0}'\frac{N!}{n_1!n_2!n_3!...}\exp{(-\beta\sum^{\infty}_{k=1}n_kE_k)}\\ &=\frac{1}{N!}\Big(\sum^{\infty}_{k=1}\exp{(-\beta E_k)}\Big)^N\\ &=\frac{1}{N!}(Z(T,V,1))^N, \end{split}$$ thereof prime on the summation means we calculate the partition function when the total number of particles is fixed, that is $$n_1+n_2+...=N.$$ $$\frac{1}{N!}$$ is Gibbs' factor. My question is why do we put a Gibb's factor here? In the precedent, we suppose the particles are distinguishable—the presence of Gibbs' factor here contradicts the assumption of distinguishable particles.

Note: Here I use canonical ensemble.

Sources:

From Thermodynamics and Statistical Mechanics by Walter Greiner, Ludwig Neise, Horst Stöcker, D. Rischke chapter 12, Grand canonical description of ideal quantum system

and

Statistical Mechanics, 4th edition (R.K. Pathria, Paul D. Beale) chapter 6.1-6.2

• Sources and references? Aug 28 at 13:01
• @TobiasFünke oh I forgot it. I found it from Thermodynamics and Statistical Mechanics by Walter Greiner, Ludwig Neise, Horst Stöcker, D. Rischke chapter 12, Grand canonical description of ideal quantum system. Also, I found the similar arguement from Statistical Mechanics, 4th edition (R.K. Pathria, Paul D. Beale) chapter 6.1-6.2 Aug 28 at 16:12
• This paper (as well as earlier ones) by Swendsen are rather interesting, in my opinion. He argues quite convincingly that the usual derivation of the $N!$ from quantum indistinguishability is, in spite of its popularity, not the correct one. The basis of his argument is that the usual definition of the entropy as $k_B$ times the volume in phase space is incorrect and does not correspond to the definition Boltzmann actually proposed (which would be $k_B$ times the probability of the macrostate). Aug 28 at 17:07
• @YvanVelenik Thank you for the reply Aug 29 at 13:41