An integral of the Planck radiation law- 2.701kT

[marcus]

an integral of the Planck radiation law-- 2.701kT

this is classical (vintage 1905) physics---kind of cute

Planck's radiation law (one of the things that kicked off QM)
can be written to show the numbers of photons at each frequency--in effect, a bar graph showing how the population of thermal photons at some temperature is distributed by freq.
It is continuous, but otherwise rather like a bar graph.

If you integrate this form of the radiation law you can learn the energy of an average photon, or equivalently the average quantum frequency.

Average quantum energy turns out to be 2.701kT
where k is Boltzmann k, and T is temp.

Average angular freq is that divided by hbar.

Now the question is, what is this number 2.701.

Turns out its an offspring of the Riemann zeta function

Look at sum of reciprocal fourth powers divided by sum of reciprocal cubes.

1+ 1/16+1/81 + 1/256 +... divided by 1 + 1/8 +1/27 + 1/64 +...

That times three is 2.701...

It is a mathematical constant like pi---which is also calculable by a series.

the Riemann zeta function is a sum of reciprocal x-th powers of the natural numbers. its good for number theory, studying primes etc. But here it is somehow related to heat. The heat glow off of warm objects. Thermal radiation.

My advice: Remember the constant 2.701 and forget the number theory. It gives a handle on the thermal radiation at any temperature.

 

[Tyger]

I've always enjoyed number theory

even made a couple of discoveries myself, nice to know there is such a direct link to a physical problem. If anyone is looking for a good description of the Blackbody Problem there is one in Abraham Pais's book about Einstein and in "The Quantum Physicists" and when I think of the author's name I'll edit it in.

Planck wanted to develop the area of chemical thermodynamics but found to his dismay that the American Josiah Willard Gibbs had completely covered the subject. Not until Berzelius included electrochemistry was anything added to the field. So he took up the unsolved problem of blackbody radiation. It's a case study in creativity and every budding physicist should read about it.

 

[jeff]

Yeah, I really enjoyed pais's technical biography of einstein.

 

[jeff]

Originally posted by marcus
Now the question is, what is this number 2.701. Turns out its an offspring of the Riemann zeta function

Actually, the real question is how precisely does the zeta function turn up here in the first place. Would you mind explaining this Marcus?

 

[marcus]

Originally posted by steinitz
Actually, the real question is how precisely does the zeta function turn up here in the first place. Would you mind explaining this Marcus?

I yield to you. You are welcome to give the explanation.

I worked thru this years ago in grad school-----it involves a series of integrals which are integration-by-parts, each step accumulating one of the 1/n³ terms.

You have presumably seen the proof more recently since you are a "young" researcher, as you said recently. So why don't you explain?

BTW did you happen to know that zeta(4) = pi⁴/90?
Answer honestly please

 

[marcus]

Originally posted by Tyger
even made a couple of discoveries myself, nice to know there is such a direct link to a physical problem.If anyone is looking for a good description of the Blackbody Problem there is one in Abraham Pais's book about Einstein and in "The Quantum Physicists" and when I think of the author's name I'll edit it in.

Planck wanted to develop the area of chemical thermodynamics but found to his dismay that the American Josiah Willard Gibbs had completely covered the subject. Not until Berzelius included electrochemistry was anything added to the field. So he took up the unsolved problem of blackbody radiation. It's a case study in creativity and evey budding physicist should read about it.

Tyger, I appreciate the Liberal Arts breadth here---your signature is, by my reckoning, also the best line out of that play. Curious as to what number theory results have occurred to you. I played around with series as a kid and found some things (but of course were already known!)

Havent read Pais book. I wonder if it has this little known fact about Planck. This is amazing and goes against the conventional picture.
His 1899 paper "On irreversible radiaton processes" (IIRC) contains a list of the Planck units, including temperature.
All of them---length, mass, time, etc---have roughly the same values we have today. I have looked at the original in
German and also examined it in translation. Can testify to this.

But Planck's constant did not appear until the 1900 paper on the black body radiation spectrum!

Yet we know, or believe we know, that the Planck units are based on hbar, G, and c. How did they emerge in Planck's mind even before the usual h constant? How did he think of them and calculate them before he even discovered the black body radiation law?

For historical richness----the 1899 paper was presented in the spring of that year in Berlin before a meeting of the (get this)
"Prussian Academy of Sciences". You can see that it was Gibbsian
in its inspiration---obsessed with thermodynamics but applying it to light itself.

I'll try to find my xerox of the paper and correct the title if I have it wrong here.

cheers

PS: ζ(4) = 1 + 1/16 + 1/81 +... = pi⁴/90

 

[jeff]

Originally posted by marcus


1. it involves a series of integrals which are integration-by-parts, each step accumulating one of the 1/n

2. So why don't you explain?

3. I worked thru this years ago in grad school.

1. Yes.

2. Asking you to do it didn't seem out of line since you began this thread.

3. I think that's great. Would you mind talking a little about what you studied and the origin of your interest in physics?

 

[Tyger]

Here's a number theory result

that I've worked out, already posted to the board. So far as I can find I'm the only one to have derived it.

https://www.physicsforums.com/showthread.php?s=&threadid=2061

Number Theory is just an occasional intertainment to me, but still it's nice to know that I solved a 150 year old riddle that has stumped some of the great minds.

 

[marcus]

Originally posted by Tyger
that I've worked out, already posted to the board. So far as I can find I'm the only one to have derived it.

https://www.physicsforums.com/showthread.php?s=&threadid=2061

Number Theory is just an occasional intertainment to me, but still it's nice to know that I solved a 150 year old riddle that has stumped some of the great minds.

I am impressed.
I looked it over but did not check details, but assuming that
it is all right then it is a very nice result.
there is a journal of recreational mathematics
did you ever see it?
I think the most efficient way to find out if what you discovered
is known is to submit it for publication
If you live close to a university
the math reference librarian at the math library would
show you some of the journals that publish
cute, nifty, but minor results.
I liked what you said about "cerulean" blue of the sky
Cerulean is a good word.
But I fail to understand how scattering from molecules does
not "significantly contribute" it. Do you?