Derivation of the Stefan-Boltzmann law...

[Orion1]

Photon species total effective degeneracy number:
[math]\boxed{N_{\gamma} = 2}[/math]

Photon radiation energy radiant emmittance Bose-Einstein distribution integration via substitution:
[math]j^{*} = \sigma_{\gamma} T_{\gamma}^4 = \int_{0}^{2 \pi} \int_{0}^{\frac{\pi}{2}} \cos \phi \sin \phi \; d \phi \; d \theta \; \frac{N_{\gamma} E_{\gamma}^4}{c^2 (2 \pi \hbar)^3} \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_{\gamma} (T_{\gamma})}} - 1} d \omega = \frac{\pi N_{\gamma} (k_B T_{\gamma})^4}{c^2 (2 \pi \hbar)^3} \left(\frac{\pi^4}{15}\right) = \frac{N_{\gamma} \pi^2 (k_B T_{\gamma})^4}{120 c^2 \hbar^3}[/math]

Radiant emmittance Stefan-Boltzmann constant:
[math]\boxed{\sigma_{\gamma} = \frac{N_{\gamma} \pi^2 k_B^4}{120 c^2 \hbar^3}}[/math]

Radiant emmittance Stefan-Boltzmann law:
[math]\boxed{j^{*} = \frac{N_{\gamma} \pi^2 (k_B T_{\gamma})^4}{120 c^2 \hbar^3}}[/math]

Photon radiation energy density Bose-Einstein distribution integration via substitution:
[math]\epsilon_{\gamma} = \alpha_{\gamma} T_{\gamma}^4 = \int_{0}^{2 \pi} \int_{0}^{\pi} \sin \theta \; d \theta \; d \phi \; \frac{N_{\gamma} E_{\gamma}^4}{(2 \pi \hbar c)^3} \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_{\gamma} (T_{\gamma})}} - 1} d \omega = \frac{4 \pi N_{\gamma} (k_B T_{\gamma})^4}{(2 \pi \hbar c)^3} \left(\frac{\pi^4}{15}\right) = \frac{N_{\gamma} \pi^2 (k_B T_{\gamma})^4}{30 (\hbar c)^3}[/math]

Photon radiation constant:
[math]\boxed{\alpha_{\gamma} = \frac{N_{\gamma} \pi^2 k_B^4}{30 (\hbar c)^3}}[/math]

Photon radiation energy density:
[math]\boxed{\epsilon_{\gamma} = \frac{N_{\gamma} \pi^2 (k_B T_{\gamma})^4}{30 (\hbar c)^3}}[/math]

Planck's law: (ref. 1)
[math]\boxed{I_{\gamma}(\nu,T_{\gamma}) = \frac{N_{\gamma} h \nu^3}{c^2 \left(e^{\frac{E_t}{E_{\gamma}}} - 1\right)}}[/math]

Radiant emmittance integration via substitution: (ref. 2)
[math]j^* = \int d\Omega \int_0^\infty I_{\gamma}(\nu,T_{\gamma}) \; d\nu[/math]

[math]\int d\Omega = \int_0^{2 \pi} \int_0^{\frac{\pi}{2}} \cos \phi \sin \phi \; d \phi \; d\theta[/math]

[math]j^* = \int_0^{2 \pi} \int_0^{\frac{\pi}{2}} \cos \phi \; \sin \phi \; d \phi \; d\theta \int_0^\infty I_{\gamma}(\nu,T_{\gamma}) \; d\nu[/math]

[math]j^* = \int_0^{2 \pi} \int_0^{\frac{\pi}{2}} \cos \phi \; \sin \phi \; d \phi \; d\theta \; \frac{N_{\gamma} h}{c^2} \int_0^\infty \frac{\nu^3}{e^{\frac{E_t}{E_{\gamma}}} - 1} \; d\nu[/math]

Differential calculus theorem:
[math]\boxed{\frac{\int_a^b f(u)^n \; du}{\int_a^b f(v)^n \; dv} = \left( \frac{du}{dv} \right)^{n+1}}[/math]

[math]\int_0^\infty \frac{\nu^3}{e^{\frac{E_t}{E_{\gamma}}} - 1} \; d\nu = \left( \frac{d\nu}{du} \right)^4 \int_0^\infty \frac{u^3}{e^u - 1} \; du[/math]

[math]j^* = \int_0^{2 \pi} \int_0^{\frac{\pi}{2}} \cos \phi \; \sin \phi \; d \phi \; d\theta \; \frac{N_{\gamma} h}{c^2} \left( \frac{d\nu}{du} \right)^4 \int_0^\infty \frac{u^3}{e^u - 1} \; du[/math]

[math]\frac{d\nu}{du} = \frac{E_{\gamma}}{h}[/math]

[math]j^* = \int_0^{2 \pi} \int_0^{\frac{\pi}{2}} \cos \phi \; \sin \phi \; d \phi \; d\theta \; \frac{N_{\gamma} h}{c^2} \left( \frac{E_{\gamma}}{h} \right)^4 \int_{0}^\infty \frac{E_t (\nu)^3}{e^{\frac{E_t (\nu)}{E_{\gamma} (T_{\gamma})}} - 1} d \nu[/math]

[math]j^* = \int_0^{2 \pi} \int_0^{\frac{\pi}{2}} \cos \phi \; \sin \phi \; d \phi \; d\theta \; \frac{N_{\gamma} E_{\gamma}^4}{c^2 h^3} \int_{0}^\infty \frac{E_t (\nu)^3}{e^{\frac{E_t(\nu)}{E_{\gamma} (T_{\gamma})}} - 1} d \nu[/math]

[math]j^{*} = \int_{0}^{2 \pi} \int_{0}^{\frac{\pi}{2}} \cos \phi \sin \phi \; d \phi \; d \theta \; \frac{N_{\gamma} E_{\gamma}^4}{c^2 (2 \pi \hbar)^3} \int_{0}^\infty \frac{E_t (\omega)^3}{e^{\frac{E_t (\omega)}{E_{\gamma} (T_{\gamma})}} - 1} d \omega = \frac{N_{\gamma} \pi^2 (k_B T_{\gamma})^4}{120 c^2 \hbar^3}[/math]

Radiant emmittance Stefan-Boltzmann law: (ref. 3)
[math]\boxed{j^{*} = \frac{N_{\gamma} \pi^2 (k_B T_{\gamma})^4}{120 c^2 \hbar^3}}[/math]

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The Stefan-Boltzmann law was first theorized in 1879, Planck's law was first theorized in 1914, I attempted to trace the derivation of the Stefan-Boltzmann law and Planck's law to determine the mathematical and theoretical origin of the number "2" in the numerator of Planck's law, however there was no formal derivation formulas published in their original papers.

According to Planck, the specific intensity[math]\; K \;[/math]of a monochromatic plane polarized ray of frequency[math]\; \nu \;[/math]is: (ref. 4, pg. 168, eq. 274, 276)
[math]K_{\nu}(\nu,T) = \frac{h \nu^3}{c^2} \frac{1}{e^{\frac{h \nu}{k_B T}} - 1}[/math]

[math]E_{\lambda}(\lambda,T) = \frac{h c^2}{\lambda^5} \frac{1}{e^{\frac{h c}{\lambda k_B T}} - 1}[/math]

Wikipedia Planck's laws: (ref. 1, ref. 5, pg. 22, eq. 1.51, 1.52)
[math]B_{\nu}(\nu,T) = \frac{2 h \nu^3}{c^2} \frac{1}{e^{\frac{h \nu}{k_B T}} - 1}[/math]

[math]B_{\lambda}(\lambda,T) = \frac{2 h c^2}{\lambda^5} \frac{1}{e^{\frac{h c}{\lambda k_B T}} - 1}[/math]

Note that the number "2" in the numerator was not published in Planck's original paper. These equations in Planck's original paper describe a scalar particle.

According to the derivation above, the number "2" in the numerator corresponds to the photon species total effective degeneracy number[math]\; N_{\gamma}[/math].

According to Rybicki and Lightman, photons have two independent polarizations (two states per wave vector)[math]\; k[/math], corresponding to the density of states (the number of states per solid angle per volume per frequency): (ref. 5, pg. 20, eq 1.47)
[math]\rho_{s} = \frac{2 \nu^2}{c^3}[/math]

Is this the theoretical definition for the total effective degeneracy number[math]\; N_{\gamma} \;[/math]for photon radiation?
[math]\boxed{\rho_{s} = \frac{N_{\gamma} \nu^2}{c^3}}[/math]

Any discussions and/or peer reviews about this specific topic thread?

Reference:
Wikipedia - Planck's law: (ref. 1)
https://en.wikipedia.org/wiki/Planck's_law

Wikipedia - Stefan–Boltzmann law - derivation from Planck's law: (ref. 2)
https://en.wikipedia.org/wiki/Stefan–Boltzmann_law#Derivation_from_Planck.27s_law

Wikipedia - Stefan-Boltzmann law: (ref. 3)
https://en.wikipedia.org/wiki/Stefan–Boltzmann_law

Planck, Maxwell (1914) - The Theory of Heat Radiation: (ref. 4)
https://archive.org/stream/theoryofheatradi00planrich?ref=ol#page/168/mode/2up

Rybicki and Lightman (1979): (ref. 5)
http://www.bartol.udel.edu/~owocki/phys633/RadProc-RybLightman.pdf

 

Edited May 14, 2018 by Orion1
source code correction...

[Mordred]

Effective degeneracy of photons.

Spin  2 two polarization states

flavor 1

color 1

particle/antiparticle 1

total degeneracy 2.

for quarks 

flavor 6

particle/antiparticle 2

color 3

spin 2 two polarization states.

total degeneracy 72.

Each group is the effective degrees of freedom, take those and multiply together to get total degeneracy. Spin is the 2 polarization states of the photon ie dipolar.

 You should end up with 118 degeracies for the elementary particles including Higgs.

Too long out of practice on the Stephen Boltzmann derivitaves but I didn't spot any mistakes in the above. Though its difficult to tell but you seem to be wondering why the degenercy is 2 for photons. (hope that assessment is correct if so there is why)

Edited May 14, 2018 by Mordred