Talk:Particle statistics

Classical statistics
The section on classical statistics is basically wrong. Particles can be, and usually are, considered indistinguishable in elementary classical statistical mechanics; if this were not the case there could be no entropy of mixing for a start. Also, the rest of the article heavily implies that the indistinguishability of particles in quantum mechanics is the only difference between quantum and classical statistics. This is wrong. I will think about how to improve the article. Sadly I have a lot of other things to do, however, and might not get around to it. Nathaniel Virgo (talk) 11:25, 24 June 2010 (UTC)

thermal distribution analogue to fermi-dirac or bose-einstein statistics? do anyons obey Pauli exclusion or not?
(this is a continuation of a discussion at Talk:Anyon)

I have raised a question whether it is appropriate to use the name "particle statistics" to classify algebraic properties of many-body quantum states (pure states).


 * There is of course a connection to statistics in that, when one considers statistical mechanics of the many-body system, the results of the system are strongly dependent on the algebraic structure of the system. In fact the symmetries are relevant even in the case when the particles are considered to be non-interacting, as seen in the celebrated Fermi-Dirac distribution or Bose-Einstein distribution.
 * Still, particles' properties are important beyond the applications of statistical mechanics. We can also consider pure quantum mechanics (pure states) where there are no statistics yet there still are important symmetries of the wave function.

On a related point there is a turn of phrase I've seen in a few places in wikipedia, which is more or less "fermions are particles that have Fermi-Dirac statistics". Besides the objection above (that particles don't intrinsically possess statistics), even in statistical mechanics this phrase is incorrect since (as I understand) Fermi Dirac statistics are derived for fermions under a very particular set of conditions (at thermal equilibium, and the fermions must not have interactions). So, lots of fermions do not follow Fermi-Dirac statistics, for example fermions in a Fermi liquid or in a superconductor. In fact the only requirement in F-D statistics is that the particles cannot occupy the same states and have no interactions when they are in different states. This means we could actually get F-D statistics for bosons as long as their interaction potential is engineered to prevent them being in the same state (I admit this is a stretch, but I mean here that effect of real interactions can overwhelm effects of exchange interactions).

Finally, for the sake of the Template:Statistical_mechanics sidebar, the topics listed ought to lead to a discussion with actual statistical results. What I mean is there should be something like a thermal distribution described in the article. --Nanite (talk) 19:25, 18 February 2014 (UTC)
 * First of all, anyons and Pauli exclusion. already answered this question, a half-year ago: for $θ ≠ 0$ there are, of course, no such states as $ψ ⊗ ψ$, but you may not make far-reaching conclusions from it. For bosons and fermions you know how to derive the Hilbert spaces of $N$-particle states from the one of 1-particle states. For anyons you don’t, and  doubt that it can be derived from purely combinatorial considerations and tensor calculus. IMHO one should consider an actual construction of the space of wave functions on the Euclidean plane or whatever that possesses the U(1) rotational symmetry. Although it isn’t a standard setting for anyons, start from the $L^{2}(S^{1})$ space – maybe you will be able understand something on this toy model. By the way, there are some researches on the quantum Hall effect where such structures with U(1)-symmetry appears (but what is familiar to me belongs to the framework of non-commutative geometry, where an emphasis is made on observables, not states). Possibly ’ll recall some of this stuff soon. Incnis Mrsi (talk) 20:43, 18 February 2014 (UTC)
 * Of particle statistics in general, we argue largely over philosophical and terminological questions. You said: particles don't intrinsically possess statistics. You substantiate it: one needs some ideal conditions for distributions prescribed by Bose–Einstein and Fermi–Dirac laws. But which quantitative laws do not need ideal conditions? Apart of quantitative laws, there are qualitative properties. think they intrinsically possess statistics, and it is another class (than interactions) of phenomena that affect their distributions. Search in Google: you’ll find many words like “weakly interacting” together with “Bose–Einstein”. Some papers you can find in the Internet assert that a statistics is a kind of interaction: it is an utterly harmful idea. These guys are poor philosophers. These are conceptually different, but compatible things. Your purism could lead to nothing: there are practically no non-interacting fermions ( disregard neutrinos where a particle statistics hardly has an actual application), but their repulsion on short distances adds little to F–D. The matter is more complicated with bosons, of course, because the effect of repulsion is contrary to the effect of B–E statistics. Incnis Mrsi (talk) 20:43, 18 February 2014 (UTC)

Assessment comment
Substituted at 02:19, 30 April 2016 (UTC)