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Questions regarding matter at densities so high that gravitational contraction is countered by the Pauli exclusion principle.

At high densities, matter will be compressed so much that the Pauli exclusion principle will act to counter any further compression. This causes the particles to form a degenerate gas (aka a Fermi gas). Degenerate gases play a vital role in stellar evolution (e.g. the He flash), brown dwarfs, very low-mass stars, white dwarfs, neutron stars, and gas giants. Degenerate gases differ from normal gases in numerous ways:

  1. Particles in a degenerate gas cannot all occupy the lowest energy levels and therefore they exert a pressure.

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In a normal (perfect) gas, particles occupy energy levels according to the Maxwell-Boltzmann distribution.

In a degenerate gas, all the lowest energy levels are filled up. The Pauli Exclusion Principle forbids two fermions from occupying the same quantum state. Therefore for each energy/momentum state there can only be two fermions (spin-up and spin-down). Every level between the lowest possible energy (the rest mass energy of a fermion) and the Fermi energy is occupied. The Fermi energy is a function of the number density of fermions in the gas. A gas can be considered "completely degenerate" if the kinetic energy at $E_F$ is $\gg kT$.

Since the particles have non-zero momentum, even if they are cooled towards zero temperature, then degenerate gases exert a degeneracy pressure.

  1. Degenerate gases will almost never expand from temperature increases

In a normal (perfect) gas, an increase in temperature increases the pressure. However, in a degenerate gas, temperature increases will lead to very little change in the degeneracy pressure until $kT$ becomes comparable with the kinetic Fermi energy.

This phenomenon allows white dwarfs and neutron stars to cool at almost constant radius; it slows the contraction of gas giant planets; it also allows explosive, runaway nuclear burning in the cores of low-mass stars during the "helium flash" and in type Ia thermonuclear supernovae.

  1. Degenerate gases have high thermal and electrical conductivity and low viscosity.

It is difficult for a fermion in a degenerate gas to lose energy in dissipative scattering interactions, since there are no free lower energy states for it to fall into. This means that the mean free paths of fermions in a degenerate gas are long. This leads to high thermal conductivity, low viscosity and, where charged fermions are involved, high electrical conductivity. In certain circumstances, degenerate gases can become superfluids or superconductors. These phenomena are found in neutron stars.

This means that degenerate gases tend to be isothermal.

  1. Degenerate gases have low specific heat capacity.

As it is difficult to extract the energy from a fermion in a degenerate gas, their specific heat capacity is very low. In gases where the electrons are degenerate but the ions are not (e.g. in a white dwarf, or the core of a star) this lowers the heat capacity of the gas with respect to a perfect gas, because the electrons do not contribute.

In a neutron star where all the particles form degenerate gases, the heat capacity becomes much smaller than the perfect gas case. This is why neutron stars cool extremely quickly.