When does the Bose-Einstein distribution function reduce to the Maxwell-Boltzmann distribution? When does Bose-Einstein distribution function reduce to Maxwell-Boltzmann distribution function in statistical mechanics?
 A: Recall the formula for the expected energy level occupation number in Fermi-Dirac & Bose-Einstein statistics:
$$\langle n \rangle = \frac{1}{e^{(\epsilon - \mu)/kT} \pm 1}$$
where $\epsilon$ represents the energy quanta per particle, $\mu$ is the chemical potential, $k$ is the Boltzmann constant and $T$ is the temperature. The $+$ in the denominator corresponds to fermions (Fermi-Dirac) while the $-$ corresponds to bosons (Bose-Einstein).
If the exponent term $(\epsilon - \mu)/kT$ is much greater than $1$, we can neglect the $1$ in the denominator and approximate:
$$\langle n \rangle \approx \frac{1}{e^{(\epsilon - \mu)/kT}} = e^{-(\epsilon - \mu)/kT} = \langle n_{MB} \rangle$$
which is precisely the expected occupation number for distinguishable (Maxwell-Boltzmann) particles.
Note that, if $\epsilon \approx 0$, then we find the approximation is accurate for states where $\mu \ll -kT$, which is equivalent to stating that the number of available single-particle states is much greater than the number of particles in the system—meaning that particles rarely attempt to occupy the same state, and the properties of state-sharing that distinguish bosons and fermions become irrelevant.
