system
Given the following post, respond as if you're commenting on Reddit:
human
When 2 fermions are affected by the Pauli Exclusion Principle, what actually happens to them that causes "repulsion" ?

For example in electron degenerate matter in a white dwarf, there is pressure holding up the mass from further collapse. But which force is producing that pressure?  Two electrons try to get into the same space closer to the core, and they are prevented by Pauli Exclusion, but what "pushed" them back?
gpt
Easier to understand if you think of a large collection of fermions in a small finite volume, rather than just 2.

When you attempt to force multiple fermions into a small space, they cannot all occupy the lowest kinetic-energy state.  Instead, they start to "stack up" in terms of kinetic energy or momentum.  Even if there is a mechanism for them to emit energy and achieve a lower momentum, the ones with high momentum can't because there's no unoccupied state for them to drop into.

If you now attempt to squeeze them into a slightly smaller volume, there are fewer available states within each narrow range of momentum, so you have to add enough energy to push some up into higher momentum states.  This is the equivalent of classical pressure, in that you must do work to reduce the volume.

By analogy, consider a large bin containing incompressible balls.  At very low density (only a few balls rolling around on the floor of the bin), there's no additional work required to compress the bin a bit.  At high density (the bin is partially filled with tightly packed balls) you have to add energy to lift some of the balls higher against gravity.  But that's only the case if there are more balls than can sit side-by-side on the floor of the bin.