Wednesday, May 30, 2012

black body in 4 dimensions

what would be a black body's flux-frequency relation in 4 spatial dimensions? in $n$ spatial dimensions? what would be equivalent relations for the Wien displacement law and the Stefan-Boltzmann law?

bonus question: what's $\mathrm{d}/\mathrm{d}x\: \mathrm{arcosh}(x)$?

1. here are my gut guesses, without much thinking and no explicit algebra on paper.
flux density: something with \nu^{n-1+1} times e^{-Boltzmannfactor}. Note: n-1 phase space factor, +1 for energy as function of \nu.
Wien displacement: essentially unchanged, still linear in T, but with a different scale factor.
Nice question!
Would be happy to see more elaborate replies.
Uli Bastian

2. $$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{Arcosh}(x)=\frac{1}{x+\sqrt{x^2-1}}(1+\frac{x}{\sqrt{x^2-1}})\\ =\frac{1}{\sqrt{x^2-1}}$$

3. well done! I hope i've got some time sometime, i'd like very much to derive the n-dimensional Wien- and Stefan-Boltzmann laws!

4. That in fact must be very simple. The n-dimensional Planck law is just the n-dimensional phase spherical shell volume times the Boltzmann factor, and then you calculate the zero of the derivative (Wien) and the integral wrt frequency (S-B) in the same way as in 3d. This is the background of my "gut guesses" above. The powers of T and \nu work out like in 3d, only the scale factors in front need a little bit more of work. This is why I didn't do it.
Uli Bastian.