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I have read that when modeling degenerate matter we treat it like a gas(high-speed particles and very few collisions) but almost all states are filled up.

First of all how do we know that collisions are unlikely?

We can treat systems as gases when particles distances are high and thus interactions are weak, collisions are unlikely; how can we justify that if for degenerate matter, like in a white dwarf, the density is so high and almost all states are full (below the Fermi energy).

We don't have an equation of state (because to derive that we first need to make assumptions about interactions between particles) so do we know this from experimental data?

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    $\begingroup$ Consider the collision between two particles in highly degenerate matter. If both particles are deep inside the Fermi surface, then all the otherwise possible collision outcomes are already occupied by other particles, and so they are forced to be unable to collide and instead have to essentially pass through each other (note that by symmetry, this is equal to reversing each other out of the collision). The bits that are left able to collide, are particles near the Fermi surface. Those contribute a vanishingly small proportion of particles, and so on average, collisions are unlikely. $\endgroup$
    – naturallyInconsistent
    Commented Sep 5 at 6:17
  • $\begingroup$ That is a very elegant answer $\endgroup$
    – Lina Jane
    Commented Sep 5 at 6:20

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Any interactions that involve changing the momentum/energy of degenerate fermions are heavily suppressed.

Consider elastic electron scattering in a degenerate gas. This might involve moving an electron from one momentum state to another with similar energy. But in a degenerate gas all such states are already occupied and so the interaction just doesn't happen. The only vacant states in the electron distribution are within $\sim k_B T$ of the Fermi energy, and since by definition, $k_B T \ll E_F$ in a degenerate gas, only the minority of potential interactions, involving electrons with energies close to $E_F$ can take place.

Such considerations also apply strongly to any interactions attempting to take energy away from an electron, since all lower energy states are filled. Or to interactions attempting to create a new electron, which cannot take place unless the new electron has energy $\geq E_F$.

This in turn leads to the properties of low heat capacity and very high thermal and electrical conductivity.

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