Fermi Function at High Temperature

[Parmenides]

Hello,

A question I can't seem to find a simple answer to is, what happens to the Fermi-Dirac distribution at T grows large? Mathematics suggests that it approaches 1/2, like it does when the energy becomes equal to the Fermi energy. Or, are we not allowed to use the F-D distribution for high temperatures and have to use the Boltzmann distribution instead?

 

[marcusl]

The transition, which is sharp for T near 0, broadens and flattens. In the extreme limit, it indeed approaches the constant 1/2. The Fermi-Dirac distribution approaches a Boltzmann distribution when the energy density is low (few particles per state) and the temperature is high.