Stefan-Boltzmann Equation question (qualitative)

[Trying2Learn]

Hi,

In the Stefan-Boltzmann equation for radiation heat transfer, there exists expected parameters of any model (area, constants, etc.). However, the temperature is raised to the fourth power.

Can someone explain why?

I get that it could just be because it has been "experimentally observed."

However, I still wonder if someone might be able to justify this law in words?

I am not interested in "why these constants: emissivity, or configuration factor, etc.)

I am only interested in:

Why the fourth power and not the fifth, or third, or second? Does the fourth power make some sort of "qualitative" or "descriptive" sense?

 

[vanhees71]

That's not too difficult. Look at it as an "ideal gas of photons". Photons are massless particles, i.e., there's no dimensionful quantity in the free electromagnetic field/Lagrangian/Hamiltonian to begin with. As far as free fields are concerned there's also no necessity to introduce a scale due to renormalization. Thus even on the quantum level there's no dimensionful quantity at hand.

Now to make things easy we use "natural units", where $\hbar$, $c$, and $k_{\text{B}}$ are set to 1. Then everything is measured in energy units like eV.

The only place where a dimensionful quantity enters the physical situation here is that you have to assume that you have a thermalized gas of free photons, i.e., the temperature is the only dimensionful quantity defining any scale. Now all kinds of thermodynamical quantities have a clearly defined dimension. E.g., energy density has the dimension "energy/volume". In our natural units lengths are measured in inverse eV. The dimension of energy density thus is $\text{eV}^4$, and the only dimensionful quantity at hand is the temperature, thus $u \propto T^4$. The same holds for pressure $P \propto T^4$. Also the energy flow (which is what's usually described by the Stefan-Boltzmann law) has a well-defined dimension: It's the "radiated power per unit area" or radiated "energy per unit time and unit area". Time has dimension 1/eV and are dimension $1/\text{eV}^2$, i.e., also the energy flow (or pointing vector) has to be $\propto T^4$.

Another interesting point is that the free radiation field obeys the "Ward Takahashi identity" of scale invariance, which is the trace of the energy-momentum tensor, ${T^{\mu}}_{\mu}=u-3P=0$, i.e., you get the equation of state $u=3P$ for free too (but here not from simple dimensional arguments but from a full application of Noether's theorem applied to scaling invariance).

 

[JTC]

That's not too difficult. Look at it as an "ideal gas of photons". Photons are massless particles, i.e., there's no dimensionful quantity in the free electromagnetic field/Lagrangian/Hamiltonian to begin with. As far as free fields are concerned there's also no necessity to introduce a scale due to renormalization. Thus even on the quantum level there's no dimensionful quantity at hand.

Now to make things easy we use "natural units", where $\hbar$, $c$, and $k_{\text{B}}$ are set to 1. Then everything is measured in energy units like eV.

The only place where a dimensionful quantity enters the physical situation here is that you have to assume that you have a thermalized gas of free photons, i.e., the temperature is the only dimensionful quantity defining any scale. Now all kinds of thermodynamical quantities have a clearly defined dimension. E.g., energy density has the dimension "energy/volume". In our natural units lengths are measured in inverse eV. The dimension of energy density thus is $\text{eV}^4$, and the only dimensionful quantity at hand is the temperature, thus $u \propto T^4$. The same holds for pressure $P \propto T^4$. Also the energy flow (which is what's usually described by the Stefan-Boltzmann law) has a well-defined dimension: It's the "radiated power per unit area" or radiated "energy per unit time and unit area". Time has dimension 1/eV and are dimension $1/\text{eV}^2$, i.e., also the energy flow (or pointing vector) has to be $\propto T^4$.

Another interesting point is that the free radiation field obeys the "Ward Takahashi identity" of scale invariance, which is the trace of the energy-momentum tensor, ${T^{\mu}}_{\mu}=u-3P=0$, i.e., you get the equation of state $u=3P$ for free too (but here not from simple dimensional arguments but from a full application of Noether's theorem applied to scaling invariance).

I believe you are getting to the core of this issue for, so forgive me, please, for following up like this. I am only doing it because I almost understand.

But your explanation is a bit over my head. Is there a chance, you might consider rephrasing this a bit more simply?

 

[vanhees71]

If you take out the final note on the trace, which is not really an answer to your question, what's not understood? I don't think that you get anything simpler than dimensional-analysis arguments in this case.

BTW, you labeled your posting as "A"...

 

[Trying2Learn]

If you take out the final note on the trace, which is not really an answer to your question, what's not understood? I don't think that you get anything simpler than dimensional-analysis arguments in this case.

BTW, you labeled your posting as "A"...

Yes, I see that now... I should not have put A - maybe I?

My confusion concerns these phrases, which are clouding my head...
thermalized gas of free photons,
The dimension of energy density thus is eV4" role="presentation">eV4, (why to the fourth power=)
time has dimension

Forget the trace part -- that I did not follow

Is there no simpler explanation?

I see you put a lot of time into yours, and I apologize for asking you to simplify, but I understand if you don't have the time.

 

[vanhees71]

I still don't understand the problem. Is it about the use of natural units? Using SI units makes everything even more complicated. Since the Stefan-Boltzmann law is about black-body radiation, i.e., electromagnetic radiation in thermal equilibrium at a temperature fixed by the temperature of the walls of the container, you have to understand this setup first to make sense of anything I say: Have a look at Wikipedia again first:

https://en.wikipedia.org/wiki/Black_body

 

[Trying2Learn]

I still don't understand the problem. Is it about the use of natural units? Using SI units makes everything even more complicated. Since the Stefan-Boltzmann law is about black-body radiation, i.e., electromagnetic radiation in thermal equilibrium at a temperature fixed by the temperature of the walls of the container, you have to understand this setup first to make sense of anything I say: Have a look at Wikipedia again first:

https://en.wikipedia.org/wiki/Black_body

I will check that.

I just found this...

https://www.quora.com/What-is-the-r...olute-temperature-in-the-Stefan-Boltzmann-law

 

[vanhees71]

Sure that's the derivation using the mentioned equation of state, using thermodynamics, but I think it's more complicated than the simple dimensional arguments, isn't it?

 

[Ibix]

I will check that.

I just found this...

https://www.quora.com/What-is-the-r...olute-temperature-in-the-Stefan-Boltzmann-law

In short, @vanhees71's argument boils down to "because $T^4$ is the only thing it could possibly be". He basically switches to a convenient system of units where all the constants are 1 and have no associated dimension (or unit, slightly less precisely) then shows that the flux has dimension of temperature to the fourth in this system. Since, as you observe, there's nothing to play with except temperature, the flux must be proportional to $T^4$, or the units don't work.

The arguments from thermodynamics or Planck's Law simply disguise the above a bit.

 

[vanhees71]

Well, nevertheless the full analysis not only provides the parametric dependence but also the quantitative law, i.e., the constant of proportionality!