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Can the Fermi velocity $v_F$ be equal to the speed of light, $c$, for the ultra-relativistic electron gas (or some exotic kind of degenerate strongly-interacting matter in neutron stars), by using Fermi-Dirac distribution function?

Note that the Fermi velocity is defined in terms of the derivative of the single-particle dispersion relation, that is affected (self-energy) by interactions.

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    $\begingroup$ You can not use F-D distribution for ultra-relativistic electrons, since Fermi distribution arises when the energy of electrons is low and everyone tries to go into the lowest energy state, then the states are assigned by Fermi selection rules and states upto $E_F$ are filled. As you start to heat plasma, more and more electrons will get the energy and go into higher energy states and soon the F-D distribution will turn into M-B distribution. $\endgroup$
    – hsinghal
    Commented Jun 14, 2016 at 18:01
  • $\begingroup$ @Rob Jeffries can you please elaborate why my comment is incorrect. I am an experimentalist hence not so rigorous in theory. $\endgroup$
    – hsinghal
    Commented Jul 18, 2016 at 14:17
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    $\begingroup$ @hsinghal I thought I had. A degenerate gas is one for which $E_F \gg kT$. If you compress the gas then $E_F$ rises. Once $E_F$ is comparable to or larger than the rest mass of the fermions involved, then they are relativistic. Once $E_F \gg mc^2$ they can be described as ultrarelativistic. Ultrarelativistic electron degeneracy exists in the core of white dwarfs and the crusts and interiors of neutron stars. Your objection might be true in the lab, since degeneracy in the lab can usually only be arranged by making things cold, rather than dense. $\endgroup$
    – ProfRob
    Commented Jul 18, 2016 at 15:04

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The Fermi velocity, the Fermi momentum and the Fermi temperature are all really just ways of rewriting the Fermi energy, $E_F$. The Fermi energy is the energy f or a gas of fermions such that at 0 temperature all states with energy below $E_F$ are filled and all states with energy above $E_F$ are empty.

Leaving aside the details of relativistic QM, a Fermi velocity of $c$ would imply an infinite Fermi Energy, or put another way imply that every state was filled. Clearly this does not happen.

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