- The pressure is independent of Temperature
- Energy deposition/heating of a degenerate gas therefore does not lead to its expansion. Rather, it leads to increase momentum of the particles in the degenerate gas.

Degeneracy is a direct result of the Pauli Exclusion Principle

We can use the uncertainty principle in combination with the concept of phase space. Phase space represents the combined position and momentum vectors of a particle in N dimensions. In our case, N=3, so each particle has an X,Y.Z spatial direction as well as X,Y,Z components of its momentum vector.

Therefore, in general

For a normal gas, the density determines the available volume in physical space and the tempearture determines the available volume in momentum space. But this is not the case for a degenerate gas. We will see that density alone determines phase space volume.

For the specific case of an electron gas, only two electrons (of opposite spin state) can occupy a specific cell in phase space.

A specific cell would have coordinates:

x,y,x

p_{x},
p_{y},
p_{z}

To avoid being in that cell, the next pair of electrons will need to acquire additional momentum to create a new cell in phase space. Its this additional momentum which leads to increased pressure.

Now let's consider the case of __non-relativistic__ degeneracy:

Let *N _{p}dp *= # of electonrs having momentum (p) in
the range p to p+dp. Consider a volume of 1 cm

The momentum volume element in this case is then
4pp^{2}dp and
*N _{p}dp * = # cells x 2 particles per cell.

If we want to find the total number of particles, then we have
to integrate over momentum space, where the maximum momentum
is *p _{o}*

The physical meaning of the above is that all cells in phase are filled
with partcles having momenta from o to *p _{o}* (known
as the Fermi momentum).

Therefore, if the volume gets smaller (as the stellar core is contracting),
*p _{o}* must increase.

We next want to calculate the pressure in this situation. Remember
that pressure is the rate of momentum transfer and this is
esentiall *p x v*

An important point to note is that even though the gas is degenerate, the statistical behavior of the particle velocities is exactly the same as in an ideal gas law. That is, a Maxwell-Boltzmann distribution pertains. In that situation we then have the following:

Now, note the following. As the core contracts, particle momementa
must increase but it can not do so without limit. The limit is when
the electron velocity ~ *c*, and in that case the gas is
fully relativistic.

In this case, we have to put in the relativistic energy in place
of mv^{2} above. That means that the term p^{2}/m goes
instead to p/m and the resultant degenerate pressure will go as
N^{4/3} not N^{5/3}.