Sunday 20 October 2019

statistical mechanics - From Fermi-Dirac to Maxwell-Boltzmann statistics


I have a little question I can't seem to find the answer to. It is as follows:


When does Fermi-Dirac statistics reduce to Maxwell-Boltzmann statistics?




Answer



The Fermi-Dirac distribution


$$ f = \frac{1}{1+e^{(E-\mu)/kT}} $$


where, $E$ is the energy, $\mu$ is the chemical potential and $kT$ is Boltzmann's constant times temperature reduces to the Boltzmann distribution


if $E \gg \mu$ then $e^{(E-\mu)/kT} \gg 1$ and so you can write $$ f \approx e^{-(E-\mu)/kT}. $$


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