Are There Distributions Different from Fermi-Dirac and Bose-Einstein?

[ndung200790]

Please teach me whether it is possible there are any distributions different from Fermi-Dirac and Bose-Einstein distributions.Because the Statistic Theorem only demontrates that integer spin particles can't obey Fermi-Dirac distribution and spin-haft particles can't obey Bose-Einstein distribution.

 

[xepma]

Yes, there is.

It's called fractional exclusion statistics (also known as Haldane exclusion statistics, after the person who came up with it). You can find the paper here:

http://prl.aps.org/abstract/PRL/v67/i8/p937_1

The idea comes down to generalizing the Pauli principle: you start off with some fixed system of N particles and an associated finite dimensional Hilbert space. By adding a particle to the system, while keeping the boundary conditions fixed, you switch from a Hilbert space with N particles to a Hilbert space with N+1 particles. The question is now: how are the dimensions of these two Hilbert space related? For bosons, nothing changes while for fermions the dimensionality drops by one. But you can also make the ansatz that the dimensionality drops by one only if you add two particles. This is called fractional exclusion statistics
.
P.S. It's been known for quite a while that the spin-statistics theorem in two and one dimensions has a far richer structure: you may have fractional spin and correspondingly fractional statistics as well. Examples include anyons which are realized in two-dimensional systems, and spinons which originate in certain spin chains. However, I should also mention that fractional exclusion statistics is not limited to these lower dimensions: you can have them in higher dimensions as well.