Wednesday, April 2, 2014

Planck, Bose and Boltzmann

the Planck-spectrum is determined by the Bose-distribution for indistinguishable integer-spin particles and not by the Boltzmann-distribution for classical distinguishable particles. can you show that the difference between the Stefan-Boltzmann-laws derived with both distributions vanishes in the limit of many dimensions? can you show this mathematically and give an intuitive argument?

1 comment:

  1. the integrals one would need to carry out are
    \begin{equation}
    \int_0^\infty\mathrm{d}x\:\frac{x^{n}}{\exp(x/T)-1}
    \end{equation}
    for the Bose-Einstein-distribution (by substituting a geometric series for $1/(q-1)$ with $q=\exp(-x/T)$) and
    \begin{equation}
    \int_0^\infty\mathrm{d}x\:x^n\exp(-x/t)
    \end{equation}
    for the Boltzmann-distribution in $n$ dimensions (by partial integration). Both integrals are proportional to $T^{n+1}$ and differ by a numerical factor of $\zeta(n+1)$, a function which rapidly approaches one for large $n$. Intuitively this can be understood in the way that for large dimensions, the integral picks up more contributions from the classical part of the photon spectrum and the true QM-result approaches the classical result.

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