# Statistical Mechanics/Density of States

## Thermal Average Number of Particles Distribution Function

$f(\epsilon )\equiv \ \langle N(\epsilon )\rangle$ Now, since this is a distribution function, we can find thermal average quantities of other thermodynamic quantities:

$\langle X\rangle =\sum _{n}f(\epsilon _{n},T,\mu )X_{n}$ Where n denotes the quantum orbital.

Now if the orbital step, n, is small, we can change the sum to an integral:

$\langle X\rangle =\int d\epsilon D(\epsilon )f(\epsilon ,T,\mu )X(\epsilon )$ Here, this $D(\epsilon )$ function is the function that mathematically allows us to transform the sum to the simplest possible integration (In the Thermal Radiation case, we will see that the D function is the Jacobian to spherical coordinate).

It is also better known as the...

## Density of States

The mathematics of the DoS were explained in the previous section, all that remains of importance is to explain its significance. Here D(ε) is the number of orbitals of energy between ε and ε + dε. We can think of it as the number of orbitals per volume in the shell taken in the integration, in other words, the density of states.