Introduction and Summary
In thermodynamics, we often inquire about the occupation of a given state of a system. This terminology derives from the quantum underpinnings that we have already discussed. We will want to be able to quickly say what the probability is of occupying a state of a system, and to be able to give an answer both relative to the occupation of other states as well as absolute.
To this end, we will need to develop what is known as the Boltzmann factor, a probabilistic measure of the relative occupation of a given state. Adding up all of these probabilities yields the ubiquitous partition function which we use at first to normalize our results and later to derive numerous other quantities. We will investigate how the Helmholtz Free Energy relates to the partition function.
We will apply these concepts to investigate the spectrum of electromagnetic radiation in a cavity. Such a spectrum is given by the Planck distribution function. We will learn that the energy density of this radiation is given by the Stefan-Boltzmann law of radiation.
We will consider the effects of the chemical potential on the probabilities of occupation of states, and come up with the Gibb's Sum. We will discuss how all of these tools are sufficient to tackle some challenging problems, such as the ideal gas.