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The difference between Fermi–Dirac function and Boltzmann approximation is that Fermi–Dirac function considers Pauli's exclusion principle but Boltzmann approximation doesn't. Why?

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Because at the time Boltzmann died, Pauli was only six years old and had not yet formulated his Exclusion Principle. Pauli was a genius but he was not that precocious!

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