Stirling's approximation in Fermi Statistics derivation

[daktari]

Hi People.

I was looking at the derivation(s) of Fermi-Dirac Statistics by means of the "most probable distribution" (I know the correct way is to use ensembles, but my point is related to this derivation) and it usually employs Lagrange multipliers and Stirling's approximation on the factorials of the ocupation numbers "n_i".

So I would say that this is not correct since, even if you assume n_i to be continuous, the value for "n_i" has to be lower than 1 because of Pauli's principle. Then to make the approximation that "log(n_i!) ~ n_i * log(n_i) - n_i" can not be right!

However it is ussualy done that way in most textbooks. What would you suggest as an alternative to derive Fermi-Dirac Statistics most probable distribution?

thanks,

 

[Cthugha]

Ehm, maybe my basic QM course has just been too long ago, but I do not remember any factorials of occupation probabilities in the derivation of the Fermi-Dirac statistics.

You have g_i states and f_i is the probability that a state with energy E_i is occupied, so in total you have g_i f_i occupied states.
Then the multiplicity is given by:

W=\Pi \frac{g_i!}{(g_i -g_i f_i)! (g_i f_i)!}

The f_i are indeed smaller than one, but there are no bare f_i! in the multiplicity.

 

[daktari]

Ehm, maybe my basic QM course has just been too long ago, but I do not remember any factorials of occupation probabilities in the derivation of the Fermi-Dirac statistics.

You have g_i states and f_i is the probability that a state with energy E_i is occupied, so in total you have g_i f_i occupied states.
Then the multiplicity is given by:

W=\Pi \frac{g_i!}{(g_i -g_i f_i)! (g_i f_i)!}

The f_i are indeed smaller than one, but there are no bare f_i! in the multiplicity.

Yes, but those g_i values are not large numbers. Just think on an electron gas, the value of g_i=2, which is not large enough to support the use of Stirling's approximation.

 

[daktari]

Yes, but those g_i values are not large numbers. Just think on an electron gas, the value of g_i=2, which is not large enough to support the use of Stirling's approximation.

BTW, I was assuming "g_i" was the degeneracy of each energy state.

 

[Cthugha]

Well, depending on the system these factors can be large (or it is at least justifiable to group nearly degenerate levels). However, there are also systems, where justifying this method is more complicated. See for example:
"on most probable distributions" PT Landsberg - Proceedings of the National Academy of Sciences, 1954

 

[daktari]

Well, depending on the system these factors can be large (or it is at least justifiable to group nearly degenerate levels). However, there are also systems, where justifying this method is more complicated. See for example:
"on most probable distributions" PT Landsberg - Proceedings of the National Academy of Sciences, 1954

Thanks. In fact I downloaded that article before posting on this forum, but the fact is that I don't find it very easy to follow. Thanks anyway!

 

[daktari]

Thanks. In fact I downloaded that article before posting on this forum, but the fact is that I don't find it very easy to follow. Thanks anyway!

Also I have just realized that my previous statement "electron gas, the value of g_i=2" is plainly wrong as that is only the degenerate factor due to spin but not the whole degeneracy factor on 6D phase space. Thanks.

 

[Count Iblis]

Wat you do is you consider M copies of the same system. In each separate system (consisting of, say, N electrons) there can only be one electron in each state.