# What is the condition for the number density of a gas to be ultra-relativistic or non-relativistic and degenerate or ideal

What is the condition for the number density ($n$) of a gas to be ultra-relativistic or non-relativistic and degenerate or ideal?

I found problems in this subject when I was reading about White-Dwarfs.

A book said $n \gg n_{QUR}$ or $n \ll n_{QUR}$ is the condition for being degenerate or ideal (classical), respectively. And $n_{QUR} = 8 \pi (kT/hc)^3$. Is this condition right? I know where $n_{QUR}$ comes from but I don't know what it really shows.

In that book it is said that $n \ll (mc/h)^3$ is the condition for being ultra-relativistic for a degenerate gas and $kT\gg mc^2$ is the condition for being ultra-relativistic for an ideal gas. Where do these equations come from? What is $mc/h$ ?

In another way, my question is mainly about this diagram and about how we can get the lines in this diagram. Credit: The Physics of Stars, A.C.Phillips

• $k$ Boltzmann constant
• $h$ Planck constant
• $c$ light speed
• $n$ is number density, not the number of particles
• $n_{QUR}$ is the quantum concentration number for ultra-relativistic particles

So first, the division between a classical gas and a quantum gas (note that all the gases in your plot are ideal - the term ideal refers to the particles being point-like and non-interacting, which is as true for the electrons in a white dwarf as those in the atmosphere of the Sun). The condition here is that for a classical gas, the density/temperature combination should be such that the phase-space (3d momentum $\times$ volume) occupied by each particle is much greater than $\hbar^3$. This ensures that there are more available quantum states than indistinguishable electrons to fill them.
The second criterion is to compare the momentum of particles with $mc$. Clearly if these are comparable then the particles are relativistic. In a classical gas, the momentum depends on the square root of temperature, as you quote. In a Fermi-Dirac quantum gas, the maximum momentum only depends on the density of particles (as $n^{1/3}$). Hence the differing criteria.