Talk:Fermi–Dirac statistics

Miscellaneous discussion
Awesome plots, Unc.hbar. My plot was thrown together quickly, with the intent of spurring someone else to make a better one. I see the plan worked in record time. Ed Sanville 14:27, 14 May 2005 (UTC)

Is the word "statistics" plural? I think so, but Florian Marquardt introduced some grammar suggesting it's singular. He also left in some of my plural forms, leaving the article sounding rather confused. I've changed it to be consistently plural. If anyone knows better please say so.

Another grammer matter: was quantum mechanics really invented  or was it perhaps  discovered  or maybe  introduced ?... just a thought

Also, F-D statistics are used for electrons in solids, not just electrons generally. In free space, you don't usually need QM, and inside atoms or molecules, you need non-statistical QM. -- Tim Starling
 * [I've inserted this comment out of chronological order.]
 * Statistics as a subject field is used with a singular verb.   --Bob K31416 (talk) 18:25, 11 January 2009 (UTC)


 * Electrons can be be found in other forms than 'in solids' or 'in free space', there are in every (non-solid/non gas) molecule also.


 * Oh really? Perhaps that's why I wrote above "inside atoms or molecules, you need non-statistical QM". I guess there could be applications for F-D statistics for electrons in molecular or atomic gases – perhaps at high temperatures – but informing me smugly that there are electrons in liquids and gases is probably not the best way to start such a discussion.
 * Sorry, I had not intention of being sarcastic or offending, I just wanted to state that solid-state physics]] is not the only field where F-D apply; i.e. F-D seems to be used in dense-matter physics, white-dwarf & neutron star are also (but not so good) example. - looxix 22:36 Mar 3, 2003 (UTC)


 * I didn't mean to suggest that F-D statistics can never be used for liquids or gases. I thought the statement "F-D statistics applies in particular to the electrons and therefore are often used in the study of solids" was misleading. It implies that anything containing electrons is suitable for study with F-D statistics. This is certainly untrue. -- Tim


 * F-D statistics is not only used for electrons in solids, in fact, a metal conductor is better interpreted as a 'gas' of electrons embedded in a solid mesh/lattice (see Fermi gas).


 * The "gas" analogy may be a useful one, but it does miss some of the more curious properties of solids. Although a small fraction of the electrons in a metal are free to move and respond to external forces (like the particles of a classical gas), not all of them are. Why not? -- Tim
 * I didn't say that all electrons are part of the "gas". - looxix 22:36 Mar 3, 2003 (UTC)


 * also F-D (as B-E) statistics apply to indistinguishable particle, if the electrons are bound in a solid, then they are distinguishable.
 * of course I may be totaly wrong. - looxix 22:14 Mar 2, 2003 (UTC)


 * About that last point, yes unfortunately you are "totaly wrong". Electrons are always indistinguishable. See identical particles for a sketchy explanation. -- Tim Starling 04:29 Mar 3, 2003 (UTC)
 * Indeed, I confounded with nuclei in a crystal. - looxix 22:36 Mar 3, 2003 (UTC)

Derivation
The new "derivation" is NOT a derivation of Fermi-Dirac statistics. The "another derivation" is much closer to a derivation, since it starts from statistical assumptions and statistical relationships. Any derivation that does not introduce the statistics of how indistinguishable particles are distributed among various energy levels while assuming each distribution is equally likely, is not a derivation. The present "derivation" introduces the partition function as a given! The partition function follows naturally from a statistical analysis of Fermions and is not a given!. The present "derivation" introduces the probability of an electron having a particular energy as a given, but this is just what a derivation is supposed to derive! This is not a derivation. It needs to go somewhere else, under a different title. By removing it I don't mean to seem like I am throwing it out, I just don't know quite where it fits. PAR 01:31, 15 May 2005 (UTC)


 * Yes, I am taking partition function for granted because in science we prove something once and then use it again and again. One can look up derivation of the partition function if (s)he so desires.  Also, someone looking up FD statistics in Wikipedia rather than SM text would more easily follow my logic than yours.  Feel free to change the title, please don't erase my work. Unc.hbar 02:08, 15 May 2005 (UTC)


 * Do you like my plots? Unc.hbar 02:21, 15 May 2005 (UTC)

I am reading and rereading what you wrote, and I cannot follow it. You say energy is 0 or epsilon, and use this, then later bring in the chemical potential from I don't know where, to fix the fact that energy is not zero. You say gi is unity, but then later say you don't need to put it in because its "put in later", implying it is not unity. When you say n-bar is a probability, you have to have a level subscript, or else its an integral, and then its not a probability. It just needs tons of work before it makes any sense. This is not to say that the second derivation is clear as a bell. It needs work too. PAR 03:14, 15 May 2005 (UTC)


 * It is a well-known fact that energy can be offset without significant implications to the problem, I don't see why I can't slip $$\mu$$ in at the last moment :). I don't like carrying the degeneracy around since it only multiplies the answer at the end.  I am not a fan of subscripts, which is odd because I study GR.


 * Statistical Mechanics is a bunch of hand-waving. There are 2 kinds of hand-waving: (1) rigorous; (2) non-rigorous.  Title my stuff as "non-rigorous hand-waving" if you will.  We could just cut and paste the book.  I think the idea if this website is to provide something books can't/don't.


 * I agree, the "derivation" is not crystal clear, I'll keep working on it.


 * I think my stuff is more undergraduate, yours is graduate-level. Maybe we should title them so?  How's FD for Dummies and FD for Hard-Core Physicists?

Also, I think the plots are excellent. I fixed the captions for mu and epsilon. I tried to do something similar with the Bose gas article. What did you use to make the plots?


 * Thanks. I used SM (Supermongo). hbar 03:47, 15 May 2005 (UTC)

-- I think that the "Derivation using Lagrange multipliers" should be changed. In particular using the Stirling's approximation for the factorial of "ni" when the quantity "ni" is a very small number (between 0 and 1 if you assume it to be a continuous variable) doesn't seem to make much sense as Stirling's approximation is good only if ni is large. 190.188.251.111 (talk) 05:55, 24 July 2009 (UTC)Muad'Dib

Dead link report
Dead link: http://www.boutichesaid.cv.dz/FermiDirac/FermiDirac.htm —Preceding unsigned comment added by Joey11 (talk • contribs) 11:27, 19 January 2009 (UTC)

Equation positioning
In the derivation that I just added, when I use a Windows based computer with non-Safari browsers, the alignment is Ok for all the equations, i.e. they are aligned by keeping the equals sign in the same place on each line. The Windows/non-Safari computer was the one that I used when I was aligning. However, when I use a MAC and Safari browser, some of the equations become misaligned. There seems to be a shifting of the lines which might be related to the use of ::: and/or hard spaces at the beginning of the line and its interaction with the TeX markup, but I'm not sure. Does anyone have any suggestions on how to get consistent alignment on both types of computers/browsers? Thanks. --Bob K31416 (talk) 11:07, 7 March 2009 (UTC)


 * I suspect you're the one responsible for the things I commented on below. If you're using something external to TeX to make it align correctly, then you're doing it wrong.  All those preceeding colons for spacing is absolutely wrong; you should be working WITHIN TeX to do that.  The "align" environment is used in TeX for alignment of "="s on different lines.  They way you (if it was you) did it is lunacy; it's a horrible mess. Michael Hardy (talk) 22:30, 23 October 2009 (UTC)

What is this mess supposed to say??

 * Rearranging the summations,
 * $$ \sum_{n_i=0} ^1 n_i \ e^{-\beta (n_i\epsilon_i)} \ \   \sideset{ }{^{(i)}}\sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$
 * $$\bar{n}_i \ = \ \frac {\qquad \qquad \qquad\qquad \qquad \qquad \qquad \qquad \quad}  { }  $$
 * $$ \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \qquad   \sideset{ }{^{(i)}}\sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$
 * $$\bar{n}_i \ = \ \frac {\qquad \qquad \qquad\qquad \qquad \qquad \qquad \qquad \quad}  { }  $$
 * $$ \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \qquad   \sideset{ }{^{(i)}}\sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$
 * $$ \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \qquad   \sideset{ }{^{(i)}}\sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$

The above is in this article and looks like something written by a lunatic. If I knew what it is supposed to say I have no doubt that I could write it correctly with much simpler TeX code than what appears here. If someone's using all those "\qquad"s in this kind of manner, that's a sure sign that they don't know how to use TeX properly. Michael Hardy (talk) 22:27, 23 October 2009 (UTC)


 * The problem was introduced with your recent edits. Please look at the version here before your recent edits and you will see that it doesn't have the problem that you introduced. Also, note that the version before your edits correctly corresponds to the way the equation appears in the cited Reif reference and is eq 9.3.2 there.


 * Re "If I knew what it is supposed to say I have no doubt that I could write it correctly with much simpler TeX code than what appears here."


 * By all means do that if you can. As I mentioned before, what it is suppose to mean is what was in the article before you edited it and introduced the problem. I would suggest that you revert the article back to what it was before you started editing and try again. --Bob K31416 (talk) 23:23, 23 October 2009 (UTC)


 * One thing I should add, in your efforts to simplify the code, do not change the appearance of the equations without consensus. --Bob K31416 (talk) 23:55, 23 October 2009 (UTC)

The problem was very much there before my edits. It said this:
 * $$ \sum_{n_i=0} ^1 n_i \ e^{-\beta (n_i\epsilon_i)} \ \   \sideset{ }{^{(i)}}\sum_{n_1,n_2,...} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$
 * $$\bar{n}_i \ = \ \frac {\qquad \qquad \qquad\qquad \qquad \qquad \qquad \qquad \quad}  { }  $$
 * $$ \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \qquad   \sideset{ }{^{(i)}}\sum_{n_1,n_2,...} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$
 * $$ \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \qquad   \sideset{ }{^{(i)}}\sum_{n_1,n_2,...} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} \;$$

I changed it to this: The former form does not just flout Wikipedia's indentation conventions (see WP:MOSMATH) but it tortures the software and other Wikipedians. It uses complicated formatting, relying both on TeX and colons, etc., outside of TeX, making it hard to edit and getting an inferior result (notice among other things how your horizontal line doesn't go as far to the right as the numerator and denominator do). That weird "\sideset" thing is nearly impossible to deal with while editing.
 * $$ \bar{n}_i = \frac{\displaystyle \sum_{n_i=0} ^1 n_i \ e^{-\beta (n_i\epsilon_i)} \sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} }{\displaystyle \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} } $$
 * $$ \bar{n}_i = \frac{\displaystyle \sum_{n_i=0} ^1 n_i \ e^{-\beta (n_i\epsilon_i)} \sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} }{\displaystyle \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} } $$

TeX is sophisticated (even the stripped-down version of it used on Wikipedia). It can do all this stuff simply, as you see above, rather than using the complicated workarounds you used. Michael Hardy (talk) 00:07, 24 October 2009 (UTC) .... The article still needs a lot of cleanup. But it's looking much better than it did six hours ago. Michael Hardy (talk) 00:09, 24 October 2009 (UTC)


 * The above latest version that you introduced contains an error. It does not include the superscript "(i)" in the two summations on the right, which is in the previous version before you started editing and the cited reference. (You may want to see the text of the article for the meaning of the superscript.) --Bob K31416 (talk) 00:22, 24 October 2009 (UTC)

OK, I've reinstated the (i).

I came to this article simply to bypass an unprintable redirect. I found all the nonstandard formatting. Usually that's easy to fix. Then I came to the place where you're using all those preceeding colons to do formatting usually done within the math tags, and they're interacting with the stuff inside the tags in an odd way. It takes a lot of close attention to things that normally wouldn't be there to figure out how to do even minor edits on a thing like that. Michael Hardy (talk) 01:33, 24 October 2009 (UTC)


 * Re "Then I came to the place where you're using all those preceeding colons to do formatting..."
 * I looked at WP:MOSMATH and I found "When displaying formulae on their own line, one should indent the line with one or more colons ..." Perhaps you had a different part in mind, since this quote doesn't prohibit what I did. Could you give the excerpt here that you were referring to? Thanks.


 * Also, I used the colons to approximately center equations and/or line up their equal signs, as is usual in most physics texts and journal articles that I recall reading. Perhaps Wikipedia's software is not sophisticated enough to do this automatically? If it can do this automatically, please implement that feature. Thanks. --Bob K31416 (talk) 04:01, 24 October 2009 (UTC)

TeX is sophisticated enough to format complex fractions. Doing it by such strange means as you used just makes things difficult to edit.

Centering is not the usage convention. It's normally indented by one colon, but if the paragraph is already intended, as by a bullet or otherwise, then two colons may be needed. Possibly more than two could be needed in some cases. Michael Hardy (talk) 01:47, 29 October 2009 (UTC)


 * I have previously and carefully simplified the equation code using \displaystyle, as indicated in the article's history. Also, see my recent additions  to Help:Displaying_a_formula which adds two examples on the subject of "Sum in a fraction".


 * Re "Centering is not the usage convention." - It is the usage convention in the scientific community outside of Wikipedia. However, left-align is the predominant usage in Wikipedia, so I have just changed the alignment in the article to left-align.
 * --Bob K31416 (talk) 14:12, 29 October 2009 (UTC)

Recent edits by Michael Hardy
Recently there were a considerable number of changes to the article. Some of these changes introduced outright physics errors, most did not improve the article, and they were accompanied by rude and untruthful remarks. These included unsubstantiated accusations of violation of Wikipedia style guidelines. If the editor insists on including these changes, consensus will be needed on this talk page through civil discussion. --Bob K31416 (talk) 12:23, 25 October 2009 (UTC)

FD distribution
The article says n_i is the "average number of fermions in the state i", which is wrong. Either an N (total number of fermions in the system) is missing in the numerator, or the given n_i is the probability that a fermion is in the state i.--190.188.10.179 (talk) 15:16, 3 October 2010 (UTC)
 * See p. 339 of the source Reif, where $$ \bar{n}_s $$ is defined for quantum distribution functions. "Let us now consider the case of arbitrary temperature T and calculate, for the several cases of interest, the mean number of particles $$ \bar{n}_s $$ in a particular state s." --Bob K31416 (talk) 13:37, 15 September 2011 (UTC)

Can someone stop Fat&Happy to revert my changes?
I made some changes at the beginning of this page. The sources are 1,2 and 5. Fermi published on March, Dirac on October of 1926 (In my country in the same year October follows March). Moreover Dirac mentioned the fact it in his book,ref. 5. Fat&Happy is continuously reverting my changed, without discussing it, and I have the opinion, without reading them. This modus operandi is everything except than collaborative... — Preceding unsigned comment added by Antonio.napoli (talk • contribs) 04:25, 22 February 2012 (UTC)

Moreover, in the Italian publication (ref. 1), the number 3 indicates the month (i.e. March). Fermi first published it in Italian, BUT some English scientists, in the same period when Dirac was going to publish his work, were already aware of Fermi work. Search for "Thomas-Fermi". Furthermore, the particles of this statistics are called Fermi's particles (by Dirac himself), the energy level, Fermi's levels, and so on and so forth....Why didn't they call Dirac's particles, or Dirac's energy if Fermi wasn't the first? Last but not the least, in science whenever a double name is assigned to a law, the common practice is to follow the alphabetic order (if the two scientists published the work contemporaneously) or the first name is usually the name of the one who defined and published the work as first.

I didn't want to change the name of the statistics, (I just wanted to specify who was the first to understand its principles), and since Dirac himself pointed this out in his book, I don't understand why this person is continuously reverting what I write.


 * If you have a reliable source that states that Fermi's work predates Dirac's, please quote it here on the talk page. Otherwise people have to dig up the source themselves and its much harder to "stop Fat&Happy". I agree with Fat&Happy that the article dates often but do not necessarily determine the order of the names. PAR (talk) 07:45, 22 February 2012 (UTC)

I mentioned the sources. Did you read what I wrote? Ref. 1, 2 and 5 of the current wikipedia page on this topic. Another valuable source, "Whereas according to Dirac, it was first studied by Fermi, and Dirac called it Fermi statistics and the corresponding particles fermions.[5]", from the same wiki page...


 * Did you read what I wrote? I asked you to quote the sources, not to just mention them. I doubt if the above quote is from "Principles of Quantum Mechanics". Find the book "Principles of Quantum mechanics" by Dirac, and quote from the book where it says that the subject was first studied by Fermi, and how Dirac called it Fermi statistics and the corresponding particles fermions. In other words, write down the sentences, here on the talk page, taken from the book, that support the above statement. PAR (talk) 14:12, 22 February 2012 (UTC)

ok, I will do, if you can explain me why I need to do this to support my points, while Fat&Happy can just delete without proving, or even mentioning anything. Where is the difference between me and this person? Which are his credits concerning the knowledge about Fermi's achievements and life to have the right to delete whatever he wants?? Following what you wrote, Wikipedia would be empty, because NOBODY reports the sentence (this article for example)...but since you desire so much to have it I will find to make you happy...You should have a look at the Italian page of Fermi, mainly written by me (Antonio Napoli), and see how many time I really reported exactly Fermi or other scientist words...exactly what you ask me, the opposite of what the large majority does... This is one example. Original letter written (but in the italian translation) from Fermi to Dirac....Use any translator to understand it. I will find also the original...because Fat&Happy can delete whatever he wants, while myself has to prove any single line... « Caro Signore! Nel suo interessante lavoro, On the theory of quantum mechanics, ha proposto una teoria del gas ideale basata sul principio di esclusione di Pauli. Ora una teoria sul gas ideale che è praticamente identica alla sua è stata pubblicata da me all'inizio del 1926 (Zs. f. Phys. 36, p. 902, Lincei Rend. Febbraio 1926). Poiché immagino lei non ha visto il mio articolo mi permetto di attrarre la sua attenzione su di esso. Sinceramente suo Enrico Fermi. »

Fermi defined the statistics before Dirac:

http://books.google.com.tw/books?id=-pL56OcVubgC&pg=PA767&lpg=PA767&dq=fermi+letter+to+Dirac+1926&source=bl&ots=jABE5WeLTD&sig=4ESQy3pPYjKXmcALr1uaPjta6eE&hl=en&sa=X&ei=YQJFT9fwG--XiAeaqcWqAw&redir_esc=y#v=onepage&q=fermi%20letter%20to%20Dirac%201926&f=false

The Creation of Wave Mechanics; Early Response and Applications 1925-1926, page 767. http://www.springer.com/physics/book/978-0-387-95180-5?cm_mmc=Google-_-Book%20Search-_-Springer-_-0

In this same page you find confirmation of what Dirac, wrote, that Fermi was the first to discover the statistics...


 * Why can't you simply find the quote and state it?


 * Looking at your first reference From Mehra and Rechenberg "The Historical Develpment of Quantum Theory" Volume 5, Part 2, I find the following quote:


 * "Paul Dirac, like Heisenberg, made use of Schroedinger's wave mechanical approach in treating quantum systems of several identical particles; without taking notice of Fermi's papers, he rediscovered Fermi (-Dirac) statistics a few months later."


 * All you have to do is state this quote, and then change the article the way you originally had it, with a reference to the above book. I am fairly sure that this will satisfy Fat&Happy. It's that simple.


 * Fat&Happy cannot delete whatever he wants. He cannot delete properly referenced statements.


 * The strict rules are that you should not enter something into Wikipedia without referencing a reliable source. If you enter something without a reliable source, then anyone can delete it. The person who deleted it does not have to prove that your unreferenced statement is wrong.


 * People who delete unreferenced statements without explanation or discussion are not being cooperative or friendly, I agree, but they are not wrong. If you know something to be true, then simply quoting the reference that proves it on the talk page will fix this. If the person now deletes it, that person is wrong.PAR (talk) 16:15, 22 February 2012 (UTC)

I guess you know you cannot copy text from google books....


 * Bye PAR (talk) 16:44, 22 February 2012 (UTC)

Average number
The first graph in this article does not show an average, but a probability. Of course, the two are linked but obviously not the same. The Physics-writer D.C. Giancoli talks about "an occupation-probability" and Hyperphysics mentions "a probability that a particle has energy E". It makes me even more suspicious about the use of an average number in this article. But correct me if I'm wrong. — Preceding unsigned comment added by Koitus~nlwiki (talk • contribs) 22:11, 11 June 2021 (UTC)
 * See Talk:Fermi–Dirac_statistics. The given source Reif derives the Fermi-Dirac Distribution as an average occupation, not a probability of occupation. However, in this case of the Fermi-Dirac Distribution  the maximum number of particles in a state is one and the average equals the probability. Compare with the Bose-Einstein Distribution where the  average occupation can be greater than 1 and thus not equal to the probability. Bob K31416 (talk) 22:55, 19 June 2021 (UTC)

Lead
The following message was moved here from my talk page.
 * "I don't understand, what's the problem with my changes? I explained the reasons in the edit summaries. Do you believe the new version is worse (in which respect)? Evgeny (talk) 15:05, 19 August 2021 (UTC)"

You started out by adding info re non-interacting particles that was already in the lead. Later you removed what was already in the lead, saying it was repetitious, and also you rewrote the lead. As I wrote in my edit summary, "please explain on the talk page what the deficiencies were that your last two edits were trying to improve and how your edits improved them...". Bob K31416 (talk) 18:01, 19 August 2021 (UTC)
 * What do you mean by "already in the lead"? A correct statement (that the FD distribution follows from the FD stats only for the case of non-interacting particles) must be made upfront. Evgeny (talk) 19:05, 19 August 2021 (UTC)
 * We could change
 * A result is the Fermi–Dirac distribution of particles over energy states.
 * to
 * A result for the case of non-interacting particles is the Fermi–Dirac distribution of those particles over energy states.
 * Bob K31416 (talk) 19:35, 19 August 2021 (UTC)
 * Fine with me, but I really see no principal difference from my version that you undid with the comment "already discussed in second paragraph". Why doesn't the second paragraph bother you now?  Evgeny (talk) 21:23, 19 August 2021 (UTC)
 * I took a second look at your version. How does this look for you.
 * Fermi-Dirac statistics is a type of quantum statistics that applies to the physics of a system consisting of many identical particles that are in thermodynamic equilibrium and obey the Pauli exclusion principle, where no two particles can occupy the same state. A result for the case of non-interacting particles is the Fermi–Dirac (F-D) distribution over energy states. F-D statistics and the F-D distribution are named after Enrico Fermi and Paul Dirac, each of whom derived the F-D distribution independently in 1926 (although Fermi derived it before Dirac).[1][2]


 * The particles of F-D statistics have half-integer spin (1/2, 3/2, etc.) and are called fermions. F–D statistics is most commonly applied to electrons, which are spin-1/2 fermions. A counterpart to F–D statistics is Bose–Einstein (B–E) statistics, which applies to identical and indistinguishable particles with integer spin (0, 1, 2, etc.) called bosons. In classical physics, Maxwell–Boltzmann (M–B) statistics applies to particles that are identical but treated as distinguishable. For both B–E and M–B statistics, more than one particle can occupy the same state, unlike F–D statistics where a state's occupancy is limited to one particle.


 * Bob K31416 (talk) 01:05, 20 August 2021 (UTC)


 * I don't know if what we're doing is right. 99% of the article's contents is about the distribution. Not much is said (and not much can be said, in fact) about the statistics. It might be better to have a short stand-alone article on quantum statistics, describing both FD and BE, and rename this article to "FD distribution". Evgeny (talk) 16:52, 20 August 2021 (UTC)
 * Maybe. My inclination is to work with what we have. Bob K31416 (talk) 20:57, 20 August 2021 (UTC)