# Grand Canonical derivation of Bose-Einstein and Maxwell-Boltzmann statistics

So, our professor introduced the Bose-Einstein statistics by deriving the Grand Canonical Partition function of a boson system associated to a single energy state $$\epsilon_r$$. So the formula is: $$\mathcal{Z} = \sum_{n = 0}^\infty \exp(-\beta(n\epsilon_r - \mu n)) = \frac{1}{1-\exp(-\beta(\epsilon_r-\mu))}$$ Then I know that we can derive the Maxwell-Boltzmann statistics of the same system (now consisting of distinguishable particles) through its Canonical Partition function. Let's assume that we have $$N$$ particles and $$m$$ different energy states. Then the formula is: $$Z = \sum_{n = 0}^N \frac{N!}{n_1!n_2!\dots n_m!} \exp(-\beta n_1 \epsilon_1)\dots\exp(-\beta n_m \epsilon_m) = (\exp(-\beta \epsilon_1)+\dots+\exp(-\beta \epsilon_m))^N$$ Now if I want to derive again the Maxwell-Boltzmann statistics, but in the Grand Canonical framework like I did for the Bose-Einstein, it does not add up. Given a single energy state $$\epsilon_r$$ (that I assume being connected to a particle reservoir with chemical potential $$\mu$$) the Grand Canonical Partition function becomes: $$\mathcal{Z} = \sum_{n = 0}^N \frac{N!}{n! (N-n)!} \exp(-\beta(n \epsilon_r - \mu n)) = (\exp(-\beta(\epsilon_r - \mu))+1)^N$$ By multiplying by the binomial coefficient I'm taking care of all possible distinguishable states (which are the number of possible ways to pick $$n$$ particles from the existing $$N$$). But when I try to evaluate the expected number of particles of this energy state, the èresult is different from the Canonical prediction. $$ = \frac{1}{\beta} \frac{\partial \log(\mathcal{Z})}{\partial \mu} = N \frac{\exp(-\beta(\epsilon_r-\mu))}{\exp(-\beta(\epsilon_r - \mu))+1}$$ and from this point I'm not able to get to the actual correct prediction: $$ = N \frac{\exp(-\beta\epsilon_r)}{\sum_r \exp(-\beta\epsilon_r)}$$ Why?

• It seems to me that Maxwell-Boltzmann statistics cannot be derived consistently from the first principles of statistical physics and it is approximation for Bose-Einstein and Fermi- Dirac statistics for the case when the average numbers of particles in one quantum state (ocupation numbers) are much less than unity. – Aleksey Druggist Jun 29 at 13:38