TITLE: What is the relationship between Maxwell–Boltzmann statistics and the grand canonical ensemble?
QUESTION [10 upvotes]: In the grand canonical ensemble one derives the expectation value $\langle \hat n_r\rangle^{\pm}$ for fermions and bosons of sort $r$:
$$ \langle \hat n_r\rangle^{\pm} \ \propto \ \frac{1}{\mathrm{exp}[(\varepsilon_r-\mu)/k_B T] \mp 1} . $$
For $(\varepsilon_r-\mu) / k_B T\gg 0$, we find 
$$ \langle \hat n_r\rangle^{\pm} \ \approx \ \frac{1}{\mathrm{exp}[(\varepsilon_r-\mu)/k_B T]} \ \propto \mathrm{exp}[-(\varepsilon_r-\mu)/k_B T].$$
The same motivation seems to be found in this Wikipedia article. However, on the same page, right at the beginning, that intuitive statement is made: 

In statistical mechanics, Maxwell–Boltzmann statistics describes the average distribution of non-interacting material particles over various energy states in thermal equilibrium, and is applicable when the temperature is high enough or the particle density is low enough to render quantum effects negligible.

Now from my derivation above, it seems that "temperature is high enough" does the opposite of helping $(\varepsilon_r-\mu) / k_B T\gg 0$ to be fulfilled. What is going on?

REPLY [0 votes]: Maxwell Boltmann statistics can be derived using canonical ensemble or microcanonical ensemble or grandcanonical ensemble. And also Fermi Dirac statistics as well as Bose Einstein statistics can also be derived from any one of the microcanonical, canonical or grandcanonical ensembles
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