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In ideal Bose and Fermi gases we often use either strongly degenerate ideal Bose/Fermi or weakly degenerate ideal Bose/Fermi gas. As far as I know mathematically if the fugacity $z=e^{\beta\mu}$ close to 1, then it is strongly degenerate Bose gas and otherwise, it is weakly degenerate Bose gas. Also, if the temperature is low then it is strongly degenerate, and if the temperature is high, then it is weakly degenerate. However, I am stuck in here since I don't have a clear picture on what does that means. Also, the degeneracy has different meaning in quantum mechanics. If possible give me some Intuitive picture into understanding strongly and weakly degenerate ideal gases

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The meaning (at least for fermion gases) is how close the occupation index is to being 1 for all energies below the Fermi energy and zero above it. This is the ideal case of a completely degenerate gas. The occupation index tells you what fraction of the available quantum states are filled as a function of the fermion energy.

In practice, this is assessed by comparing the ratio $E_F/kT$ (or if you are an astrophysicist and care about relativistic fermions $(E_F -mc^2)/kT$) with unity. If it is much bigger than 1, then the gas is strongly degenerate. If it is somewhere between 1 and 10 I would call that partially (or weakly) degenerate, but if it is less than 1 I would call that non-degenerate.

This "degeneracy parameter" is affected by both density and temperature. By definition, the numerator, the Fermi kinetic energy, only depends on the fermion number density; whereas the denominator obviously depends on temperature. Thus there are two ways you can get a degenerate gas with a degeneracy parameter much larger than unity - by decreasing the temperature or increasing the fermion density.

To get a visualisation, try this applet I wrote, which shows how the occupation index of the fermions as a function of energy changes as the degeneracy parameter changes. It is valid for strongly degenerate and weakly (or partially) degenerate fermion gases with degeneracy parameters $>1$. The applet was for use in astrophysics, so the energy includes the rest-mass energy of the fermions.

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