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For example, when an electron degenerate gas forms, two electrons (of opposite spins) occupy each of the lowest possible energy states up to the Fermi energy. This is because of the Pauli exclusion principle.

However, as all the electrons reach low energy states, where does all their energy go? If conservation of energy is going to hold up, they cannot magically lower in energy without the energy going somewhere, right? Is it emitted as photons?

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Any quantised system has a ground state and excited states, and in any quantised system relaxing into the ground state requires energy to be shed in some fashion.

In an isolated system like a hydrogen atom the energy is normally emitted as photons. However add other hydrogen atoms and this opens new routes for energy to be lost. For example an excited hydrogen atom can lose energy by colliding with another hydrogen atom and converting the energy into kinetic energy. This is the mechanism behind collision broadening.

So in your Fermi gas system there are likely to be various mechanisms for the energy to be lost, and it isn't possible to answer your question in general as it will depend on the details of the system. Emission of photons is always a mechanism, but for example your Fermi gas could also cool by evaporation i.e. energy could be transferred into kinetic energy of electrons at the edges so those electrons leave the system and carry away energy with them.

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