Hello,

I have a few questions concerning the Bose Einstein Distribution. I
recently came across the following link:
http://mathworld.wolfram.com/Bose-EinsteinDistribution.html

I couldn't help but notice that in (1) there is a k^s term in the
numerator. From what I've seen before, i've never encountered this
k^s. Let me be more specific. Gamma(k) should be the energy
distribution in this formula. When we take for example the Boltzmann
blackbody radiation law in the following link:

http://scienceworld.wolfram.com/physics/Stefan-BoltzmannLaw.html

We can see that toward the end, the integeral in (8) developes a u^3
term. While I won't quote it specifically, I believe in Fermi
statistics one gets a square term. At least from my understanding,
this is some kind of a dimension identifier related to momentum. So my
first question boils down to what exactly does the k^s represent in
the general Bose Einstein distribution I quoted earlier. Specifically,
what does k and s represent (some sort of degeneracy maybe?).
Secondly, if we were to make s a complex number of some form a+bi,
would this make any physical sense?

Thank you very much for your time. I eagerly wait for your answer!

---

Clarification of Question by ivanyakov-ga on 20 Oct 2005 15:20 PDT

I did some more research and found that there is a generalization to
dN(e)/de = A*e^s, which specifies the number of quantum states in a
range de (e being energy). So I'm starting to see why we have the k^s
term now. The term "s" seems to be found through Schroedingers
equation by comparing the energies. However i'm still rather curious
as to whether or not s can attain practically any value (and not the
usual 1/2 for nonrelativistic particles, and 2 for relativistic
particles).

K term on the first equations corresponds to the energy of the sistem,
this gamma isnīt the gamma distribution function, is the partition
function of energy on cannonical ensemble

That makes sense from what i've seen in some books, but I still can't
figure out what "s" relates to. Perhaps it has something to do with
spin? I understand that density of states generally is proportional to
a power distribution such as this k^s, but I guess i'm hoping s isn't
experimentally determined and can be derived for any particle (such as
a boson)

I think the problem is that this is a math page, not a physics page. 
While their equation is probably correct with some definition of
k,s,and mu, those values don't really have physical significance.  It
is just convienent for explicitly solving in terms of the equations
they have listed!

The more commonly seen formula can be found over on scienceworld's B-E page:
http://scienceworld.wolfram.com/physics/Bose-EinsteinDistribution.html
Or, if you prefer the probability functions (integrated, &
normalized), see hyperphysics
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/disfcn.html#c1.

You may want to try a good thermal physics book like Kitell, which
should have the derivations of all that good stuff.