Friday, September 30, 2016

Why are quantum gases called degenerate?

In my recent tutorial on bad metals at IISER Pune a student asked me a basic question that I could not answer:
"Why is the degenerate Fermi gas called "degenerate"?
Is it anything to do with degenerate energy levels?"

So, I went in search for answers.

The Wikipedia entry on Degenerate matter is a bit rambling and I found it unhelpful.

I then went to the library and looked at a few textbooks and found a range of answers. Some books use the term "degenerate" without any elaboration.

In the discussion below it seems looking at the Oxford dictionary is helpful:
Having lost the physical, mental, or moral qualities considered normal and desirable; showing evidence of decline. 
technical: Lacking some usual or expected property or quality, in particular.
Here are a few entries
degenerate. (This use of the word is completely unrelated to its other use to describe a set of quantum states that have the same energy).
Daniel V. Schroeder, An Introduction to Thermal Physics, page 272.
[my favourite undergraduate text on statistical mechanics]
... a bit of terminology. At low temperature, quantum ideal gases behave very differently from the way a classical ideal gas behaves. ..... The quantum gases are said to be degenerate at low temperatures. This is not a moral judgement. Rather, the word "degenerate" is used in the sense of departing markedly from the properties of an "ordinary" classical gas.
Ralph Baierlein, Thermal Physics, page 192.
Gas degeneration proper. The quantitative study of the deviations from the classical gas laws when xi is not very small ...
Erwin Schrodinger, Statistical Thermodynamics (1936)

Here xi is the product of the particle density and the thermal deBroglie wave length cubed.

The definition of a quantum gas is one where xi becomes larger than one. (This occurs for "high" densities and "low" temperatures).


But then there is an interpretation in terms of degeneracy of energy levels because one can consider the case where each energy level has degeneracy g and condition for a non-degenerate gas (i.e. Maxwell-Boltzmann statistics to apply) is
g >> n_i ~ exp ((mu-Ei)/kB T) = number of particles in level i

I welcome comments.

3 comments:

  1. I am not a physics person. Your blog is very good. The thesis below may be helpful. An extract below may be helpful.

    http://web.mit.edu/8.322/Spring%202007/notes/DFSCropped.pdf

    At high temperatures, particles have lots of energy and (as we shall see) many
    quantum states available to them. On the average, the probability that any quantum state is occupied is rather small (<< 1) and the exclusion principle plays little role.
    At lower temperatures, particles have less energy, fewer quantum states are available and average occupation number of each state increases. Then the exclusion principle
    becomes essential: the available levels up to some maximum energy (determined by the density) are, on average, nearly filled; higher levels are, on average, nearly empty.
    Such systems are then termed “degenerate,” hence the title of this section. Actually these statements are strictly true only at zero temperature and when the mutual interactions of the fermions are ignored.

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  2. Ross, that's a very good question :-). At some stage of my teaching 3rd year Statistical Mechanics I asked it to myself, and came up with the following answer: it's to do with occupancies of single-particle orbitals. At high T (Boltzmann distribution), the average occupancies of single-particle orbitals, or the probability of an orbital (even the lowest energy one) to be occupied, are so low (<<1) that multiple occupancies can be totally ignored. (This also leads to the fact that the only thing that needs to be done to "fix" the classical counting is to introduce the Gibbs correction factor of 1/N! and nothing else!) At low temperatures, on the other hand, multiple occupancies (at least for bosons) become more likely and therefore particle statistics needs to be taken into account (alternatively, average occupancies of lowest orbitals are no longer <<1). So, a short answer is that it's to do with multiple occupancies of orbitals.

    I just checked Huang's Statistical Mechanics, and he says this: "In this domain [T<<T_F] the gas is said to be degenerate because the particles tend to go to the lowest energy levels possible. For this reason T_F is also called the degeneracy temperature".

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  3. ... So, just like with "degenerate" energy levels, when one has multiple single-particle orbitals of the same energy, now we have multiple particles wanting to occupy the same orbital.

    Ha-ha, but reading Huang's definition again, it can be interpreted as a moral judgement: particles tend to go as low as possible.

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