Derivation of the Planck Distribution function

It's about to get real, time for some fun ha ha.
The Planck distribution function: 

This function can be used to describe the emission spectra of stars and is used to derive Planck's Law and Stefan-Boltzmann law, which I will be doing in a later blog post. It was the first application of Quantum Thermal physics. It is very elegant in that it arose from the simple idea of the quantization of energy.

A "mode" is a particular state in which a system can exist. For now, it will describe a particular oscillation pattern in a cavity, or frequency 

As we know now, the energy of the mode is given be the quantized number of photons (s) within the mode 

It is useful now to introduce the Partition function, which is the sum over all states of the Boltzmann factor:

Where tau is the fundamental temperature of the system and is equivalent to the Boltzmann constant times the Absolute temperature in Kelvin.

However, this summation is of the form:

where x is

 and so since we know that this summation converges, we can find that the value of Z is:

The probability of finding the system in a given state is:

With this we can find <s>, the thermal average value of the number of photons in a given state.

Using the substitution of:

We can find that:

and using the same trick we did with the partition function this is then:

Plugging this summation back into our thermal average value equation:

Which is equivalent to:

This is the more conventional form of the Planck Distribution Function and is applicable to any wave field with energy of the form :

Since we now know that the thermal average value of photons in a single mode of a system is given by Planck's distribution function, we can now find the thermal average energy in each mode:

Sources: Thermal Physics 2nd edition by Charles Kittel and Herbert Kroemer.