So, I’ve given you a bit of information in these past BMC posts about what statistical mechanics and thermodynamics are all about. These have been all part of the so-called microcanonical ensemble. All of this focuses around examining the individual states of each component of a thermodynamic system (the microstates). The two assumptions of the microcanonical ensemble is that the total number of particles in a system stays fixed, and that the total energy of the system is also fixed. Removing this energy requirement allows us to move into another thermodynamic formulation: the canonical ensemble. This allows us to consider things like a warming bottle of beer as just the bottle of beer, and not the bottle and the environment it is situated in (a much more complicated problem). In order to work within the canonical ensemble, a powerful tool for describing the system is needed. This tool is the Partition Function.
I’m not going to go into the derivation of the partition function, as that requires more math than I am willing to lose marks for (we aren’t supposed to use any math in these previews). Suffice is to say, it is determined by trying to find the properties of each microstate. The partition function itself is simply the sum of each microstate’s properties. It encodes within it the distribution of energy across all of the microstates in the system at a given temperature as an exponential function of the energy at the specific microstate. In so doing, it tells you essentially all there is to know about the thermodynamic “structure” of a system. By encoding the distribution of energy across the total number of microstates, it is trivial to calculate the entropy from the system (it is a simple logarithmic relationship). For example, the entropy of the system is maximized when the partition function itself is maximized, and minimized when the partition function is closest to 1. These positions correspond to energy being concentrated in one microstate (entropy is maximized) and energy being evenly distributed across all microstates (entropy is minimized). This exactly corresponds to what we have learned up until now about entropy. The partition function is a simply mathematical tool that acts like a thermodynamic “blueprint” of a system in the canonical ensemble. With it, the whole statistically mechanical world is your thermodynamic oyster.