Page:Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt.pdf/12

 Comparing this with the general formula that expresses Boltzmann's principle


 * $$S - S_0 = \frac{R}{N}\lg W,$$

we arrive at the following conclusion:

If monochromatic radiation of frequency &nu; and energy E is enclosed (by reflecting walls) in the volume v0, then the probability that at an arbitrary point in time all of the radiation energy located in a part v of the volume v0 is:


 * $$ W = {\left(\tfrac{v}{v_0}\right)}^{\tfrac{N}{R}\tfrac{E}{\beta\nu}} \ .$$

Subsequently we conclude:

In terms of heat theory monochromatic radiation of low density (within the realm of validity of Wien's radiation formula) behaves as if it consisted of independent energy quanta of the magnitude R&beta;&nu;/N.

We also want to compare the average magnitude of the energy quanta of the "black body radiation" with the mean average energy of the center-of-mass-motion of a molecule at the same temperature. The latter is 3/2(R/N)T, and for the average energy of the Energy quanta Wien's formula gives:


 * $$\frac{\int\limits_{0}^{\infty} \alpha\nu^3e^{-\frac{\beta\nu}{T}}d\nu }{\int\limits_{0}^{\infty} \frac{N}{R\beta\nu} \alpha\nu^3e^{-\tfrac{\beta\nu}{T}}d\nu } = 3 \frac{R}{N}T.$$

The fact that monochromatic radiation (of sufficiently low density) behaves as regards to dependency of entropy on volume like a discontinuous medium that consists of energy quanta of magnitude R&beta;&nu;/N suggests we should investigate whether the laws of