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

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:

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

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

↑ M. Planck, Ann. d. Phys. 1 p.99. 1900.
↑ 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
$Z=\sum \limits _{\nu =1}^{\nu =\infty }A_{\nu }sin\left(2\pi \nu {\frac {t}{T}}+\alpha _{\nu }\right)\ ,$
where $A_{\nu }\geqq 0$ and $0\leqq a_{\nu }\leqq 2\pi$ . Performing this expansion arbitrarily often with arbitrarily chosen initial times yields a range of different combinations for the quantities A_ν and α_ν. Then for the frequencies of the different combinations of the quantities A_ν and α_ν there are the (statistical) probabilities dW of the form:
$dW=f(A_{1}\ A_{2},\dots \alpha _{1}\ \alpha _{2}\dots )dA_{1}dA_{2},\dots d\alpha _{1}\ d\alpha _{2}\dots$
The radiation is then as unordered as imaginable, if
$f(A_{1},A_{2}\ \dots \alpha _{1},\alpha _{2}\dots )=F_{1}(A_{1})F_{2}(A_{2})\dots f_{1}(\alpha _{1}).f_{2}(\alpha _{2})\dots$
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 A_ν and α_ν 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.

[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
$Z=\sum \limits _{\nu =1}^{\nu =\infty }A_{\nu }sin\left(2\pi \nu {\frac {t}{T}}+\alpha _{\nu }\right)\ ,$
where $A_{\nu }\geqq 0$ and $0\leqq a_{\nu }\leqq 2\pi$ . Performing this expansion arbitrarily often with arbitrarily chosen initial times yields a range of different combinations for the quantities A_ν and α_ν. Then for the frequencies of the different combinations of the quantities A_ν and α_ν there are the (statistical) probabilities dW of the form:
$dW=f(A_{1}\ A_{2},\dots \alpha _{1}\ \alpha _{2}\dots )dA_{1}dA_{2},\dots d\alpha _{1}\ d\alpha _{2}\dots$
The radiation is then as unordered as imaginable, if
$f(A_{1},A_{2}\ \dots \alpha _{1},\alpha _{2}\dots )=F_{1}(A_{1})F_{2}(A_{2})\dots f_{1}(\alpha _{1}).f_{2}(\alpha _{2})\dots$
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 A_ν and α_ν 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.

[1]

[2]

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

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