# Statistical Mechanics/Thermal Radiation

## Contents

## Planck Distribution Function[edit]

For thermal radiation we know the following equation:

ε_{n}=sℏω_{n}

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

Z = Σ_{s=0} exp(-sℏω_{n}/T)

By algebra:

= 1/(1 - exp(-ℏω_{n}/T))

Therefore, we can also find the probability:

P(s) = exp(-sℏω_{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:

<s> = Σ_{s=0} sP(s) = Z^{-1} Σsexp(-sℏω_{n}/T)

Which if we carry out the mathematics of the sum:

<s>=1/(exp(ℏω_{n}/T) - 1)

## Stefan-Boltzmann Law[edit]

Remember that for a mode:

ε_{n}=sℏω_{n}

Average it:

<ε_{n}> = <sℏω_{n}>

= <s>ℏω_{s}

From the previous section:

= ℏω_{n}/(exp(ℏω_{n}/T) - 1)

Thus, if we sum up over all the modes:

U = Σ_{n} ℏω_{n}/(exp(ℏω_{n}/T) - 1)

Note that ω_{n} = nπc/L, now because ℏ is so small, we can approxiamate 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 = πℏcn/LT (an extra 1/8 comes in becaues 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) = 4n ^{2} because of the spherical shell * 1/8 * 2 = n^{2}, ε=ℏω_{n}, and f(ε)=(exp(ℏω_{n}/T) - 1)^{-1}*

U = (L^{3}T^{4}/π^{2}ℏ^{3}c^{3}) ∫_{0}^{∞} x^{3}/(exp(x) - 1) dx

The integral has a definite value found in an integral table, L^{3}=V, and thus we come upon the stefan-Boltzmann law of radiation:

U/V = π^{2}/15π^{2}ℏ^{3}c^{3} T^{4}

## Planck Radiation Law[edit]

Now, in our previous derivation, instead of integrating in terms of dn, say we left it as dω, there would be something of the form:

U/V = ∫dω u_{ω}

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_{ω} = ℏ/π^{2}c^{3} ω^{3}/(exp(ℏω/T) - 1)

And this is known as Planck's radiation Law.

## Kirchoff's Law[edit]

Say we are concerned with the radiant flux density, by the definition of flux density:

J_{U} = cU(T)/4V (the extra 4 is a geometrical factor)

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

J_{U} = π^{2}T^{4}/60ℏ^{3}c^{2}

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