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marcus

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**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.

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