Mathematically, the Fermi-Dirac (FD) distribution and Bose-Einstein (BE) distribution coincides with the Maxwell-Boltzmann (MB) distribution in the limit $(\epsilon-\mu)/k_BT\gg 1$. Therefore, in this limit the “quantum nature” of the particles i.e., the indistinguishability must be lost. What is physically happening inside the system in this limit and away from it?
Roughly I can understand that if the indistinguishability is lost then the counting of microstates becomes classical. The comparison of interparticle separation and thermal de Broglie wavelength reveals that the quantum mechanical nature of a quantum gas is lost at high temperature. However, this is apparently in contradiction with the limit $(\epsilon-\mu)/k_BT\gg 1$ at which a quantum gas becomes classical.e., the BE and FD distributions go over to MB distribution. This limit says for a quantum gas to behave classically, the temperature has to be low! But this is opposite to what the case usually is-a gas behaves quantum mechanically at low temperatures.