# Page:Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt.pdf/4

and gas molecules will return its average energy to ${\displaystyle {\bar {E}}}$ by absorbing or releasing energy. Hence, in this situation, dynamic equilibrium can only exist when every resonator has an average energy ${\displaystyle {\bar {E}}}$.

We apply a similar consideration now to the interaction between the resonators and the ambient radiation within the cavity. For this case, Planck has derived the necessary condition for dynamic equilibrium [1]; treating the radiation as a completely random process.[2]

He found:

${\displaystyle {\bar {E}}_{\nu }={\frac {L^{3}}{8\pi \nu ^{2}}}\rho _{\nu }.}$

Here, ${\displaystyle {\bar {E}}_{\nu }}$ is the average energy of a resonator of eigenfrequency ν (per oscillatory component), L is the speed of light, ν is the frequency, and ρν is the energy density of the cavity radiation of frequency between ν and ν + .

1. M. Planck, Ann. d. Phys. 1 p.99. 1900.
2. This condition can be formulated as follows. We expand the Z-component of the electric force (Z) in a given point in the space between the time coordinates of t=0 and t=T (where T is a large amount of time compared to all the vibration periods considered) in a Fourier series
${\displaystyle Z=\sum \limits _{\nu =1}^{\nu =\infty }A_{\nu }sin\left(2\pi \nu {\frac {t}{T}}+\alpha _{\nu }\right)\ ,}$
where ${\displaystyle A_{\nu }\geqq 0}$ and ${\displaystyle 0\leqq a_{\nu }\leqq 2\pi }$. Performing this expansion arbitrarily often with arbitrarily chosen initial times yields a range of different combinations for the quantities Aν and αν. Then for the frequencies of the different combinations of the quantities Aν and αν there are the (statistical) probabilities dW of the form:
${\displaystyle dW=f(A_{1}\ A_{2},\dots \alpha _{1}\ \alpha _{2}\dots )dA_{1}dA_{2},\dots d\alpha _{1}\ d\alpha _{2}\dots }$
The radiation is then as unordered as imaginable, if
${\displaystyle f(A_{1},A_{2}\ \dots \alpha _{1},\alpha _{2}\dots )=F_{1}(A_{1})F_{2}(A_{2})\dots f_{1}(\alpha _{1}).f_{2}(\alpha _{2})\dots }$
That is if the probability of a particular value of A and α respectively is independent of the value of other values of A and x respectively. The more closely the demand is satisfied that the separate pairs of values Aν and αν depend on the emission and absorption process of separate resonators, the more closely will the examined case be one of being as unordered as imaginable.