If I have a beaker that is at zero temperature in thermal equilibrium with its surrounding. If I start filling fermions (say electrons) in it, then according to the Fermi-Dirac statistics, the energy of the beaker after I have filled it with $N$ electrons will be related to the number of particles and the volume of the beaker, I am confused in which form is the Fermi energy stored in the electrons? If it its in form of translational kinetic energy, then wouldn't those electrons strike the beaker at high speeds and increase the temperature of the beaker from 0 kelvin to some finite amount? I am a beginner and I think I am missing something here.
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Related: https://physics.stackexchange.com/a/549647/247642 – Roger V. Jul 04 '22 at 11:49
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A beaker of electrons at zero temperature is counterfactual. When given a problem like this in physics class, you're supposed to close your eyes to reality and consider only the mathematical abstraction that you're reasoning about. – John Doty Jul 04 '22 at 13:02