Statistical Mechanics/Thermal Radiation

Planck Distribution Function
For thermal radiation we know the following equation:

&epsilon;n=s&hbar;&omega;n

which we can apply our previously made Partition-function 'Infrastructure' to:

Z = &Sigma;undefined exp(-s&hbar;&omega;n/T)

By algebra:

= 1/(1 - exp(-&hbar;&omega;n/T))

Therefore, we can also find the probability:

P(s) = exp(-s&hbar;&omega;n/T)/Z

Now, we can start calculating some interesting thermodynamic quantities. Let's start with the thermal average of s, the average mode of thermal radiation given a certain temperature:

&lt;s&gt; = &Sigma;undefined sP(s) = Z-1 &Sigma;sexp(-s&hbar;&omega;n/T)

Which if we carry out the mathematics of the sum:

&lt;s&gt;=1/(exp(&hbar;&omega;n/T) - 1)

Stefan-Boltzmann Law
Remember that for a mode:

&epsilon;n=s&hbar;&omega;n

Average it:

&lt;&epsilon;n&gt; = &lt;s&hbar;&omega;n&gt;

= &lt;s&gt;&hbar;&omega;s

From the previous section:

= &hbar;&omega;n/(exp(&hbar;&omega;n/T) - 1)

Thus, if we sum up over all the modes:

U = &Sigma;n &hbar;&omega;n/(exp(&hbar;&omega;n/T) - 1)

Note that &omega;n = n&pi;c/L, now because &hbar; is so small, we can approximate this sum to an integral. In the process we will change the coordinates of the integral over n in spherical coordinates, and we will let x = &pi;&hbar;cn/LT (an extra 1/8 comes in because we are integrating over only positive values of n, and an extra 2 due to two independent set of cavity modes of frequencies):

Note: actually, this is a density of states problem with D(n) = 4n2 because of the spherical shell * 1/8 * 2 = n2, &epsilon;=&hbar;&omega;n, and f(&epsilon;)=(exp(&hbar;&omega;n/T) - 1)-1

U = (L3T4/&pi;2&hbar;3c3) &int;0&infin; x3/(exp(x) - 1) dx

The integral has a definite value found in an integral table, L3=V, and thus we come upon the Stefan-Boltzmann law of radiation:

U/V = &pi;2/15&pi;2&hbar;3c3 T4

Planck Radiation Law
Now, in our previous derivation, instead of integrating in terms of dn, say we left it as d&omega;, there would be something of the form:

U/V = &int;d&omega; u&omega;

Carrying along the comparison with statistical properties, this is like a density, to be specific, a spectral density, if we carry the math out:

u&omega; = &hbar;/&pi;2c3 &omega;3/(exp(&hbar;&omega;/T) - 1)

And this is known as Planck's radiation Law.

Kirchhoff's Law
Say we are concerned with the radiant flux density, by the definition of flux density:

JU = cU(T)/4V (the extra 4 is a geometrical factor)

If we take and apply the Stefan-Boltzmann law to this:

JU = &pi;2T4/60&hbar;3c2

The only difference between this and Kirchhoff's law is an extra constant thrown in known as the absorption/emissivity constant, dependent on the material.