On a Heuristic Point of View Concerning the Generation and Transformation of Light

§ 4. Limiting law for the entropy of monochromatic radiation at low radiation density.Show the German source

It does follow, to be sure, from the observations of “black radiation” so far that the law originally set up by Mr. W. Wien for “black radiation”Show the German source $$\varrho = \alpha \nu^3 e^{-\beta \frac{\nu}{T}}$$$\varrho$Spectral energy density$\alpha$Spectral constant A$\nu$Frequency of the light$\beta$Spectral constant B$T$Temperature of the radiationExplain this equationShow the German source is not exactly valid.Show the German source But for large values of $\nu/T$ it has been confirmed very completely by experiment.Show the German source We take this formula as the basis of our calculations, but bear in mind that our results hold only within certain limits.Show the German source

From this formula there follows first:Show the German source $$\frac{1}{T} = -\frac{1}{\beta \nu} \lg \frac{\varrho}{\alpha \nu^3}$$$T$Temperature of the radiation$\beta$Spectral constant B$\nu$Frequency of the light$\varrho$Spectral energy density$\alpha$Spectral constant AExplain this equationShow the German source and further, using the relation found in the preceding section:Show the German source $$\varphi(\varrho, \nu) = -\frac{\varrho}{\beta \nu} \left\{ \lg \frac{\varrho}{\alpha \nu^3} - 1 \right\}.$$$\phi$Spectral entropy density$\varrho$Spectral energy density$\nu$Frequency of the light$\beta$Spectral constant B$\alpha$Spectral constant AExplain this equationShow the German source Now let there be given a radiation of energy $E$ whose frequency lies between $\nu$ and $\nu + d \nu$.Show the German source Let the radiation occupy the volume $v$.Show the German source The entropy of this radiation is:Show the German source $$S = v \varphi(\varrho, \nu) d \nu = -\frac{E}{\beta \nu} \left\{ \lg \frac{E}{v \alpha \nu^3 d \nu} - 1 \right\}.$$$S$Radiation entropy$v$Volume$\phi$Spectral entropy density$\varrho$Spectral energy density$\nu$Frequency of the light$E$Radiation energy in the band$\beta$Spectral constant B$\alpha$Spectral constant AExplain this equationShow the German source If we confine ourselves to investigating the dependence of the entropy on the volume occupied by the radiation, and denote by $S_0$ the entropy of the radiation when it has the volume $v_0$, we obtain:Show the German source $$S - S_0 = \frac{E}{\beta \nu} \lg \left( \frac{v}{v_0} \right).$$$S$Radiation entropy$S_0$Radiation entropy$E$Radiation energy in the band$\beta$Spectral constant B$\nu$Frequency of the light$v$Volume$v_0$VolumeExplain this equationShow the German source

This equation shows that the entropy of a monochromatic radiation of sufficiently small density varies with the volume according to the same law as the entropy of an ideal gas or that of a dilute solution.Show the German source The equation just found will be interpreted in what follows on the basis of the principle introduced into physics by Mr. Boltzmann, according to which the entropy of a system is a function of the probability of its state.Show the German source

Bern, 17 March 1905.Show the German source

(Received 18 March 1905.)Show the German source