Quantum Occupation Distributions

Three rules for how particles fill the available energy states, and the difference between them is the whole content of quantum statistics. Identical fermions obey the exclusion principle, so a state holds at most one and the occupation is the Fermi-Dirac function $n=1/(e^{(E-\mu)/kT}+1)$, never exceeding one and equal to exactly one half at the chemical potential $\mu$. Identical bosons have no such limit and crowd into the lowest states; the Bose-Einstein occupation $n=1/(e^{(E-\mu)/kT}-1)$ diverges as $E\to\mu$ and is defined only above it. Distinguishable classical particles follow Maxwell-Boltzmann, $n=e^{-(E-\mu)/kT}$, the dilute limit both quantum laws collapse to once states are sparsely filled. Drag the energy cursor to read all three occupations, move the chemical potential, and let the temperature sweep: at low $kT$ the Fermi-Dirac curve sharpens into a step at $\mu$, the boundary between filled and empty states that defines the Fermi sea, while the Bose-Einstein curve piles up ever more steeply just above $\mu$. The lower panel shows the same three curves on a logarithmic axis, where the high-energy tails straighten into a single line: the regime $E-\mu\gg kT$ in which quantum statistics is invisible and the classical exponential is exact.