In our discussion so far we described the canonical ensemble of N identical particles or molecules. We found that from the canonical partition sum we can recover the free energy which is one of the thermodynamic potentials introduced in the first chapter. A natural question is whether there are other approaches to statistical mechanics which are in turn related to other state functions such as the entropy. In this chapter we will see that this is indeed the case. We will end up with the complete picture of how different probability measures in statistical mechanics are related to the various potentials in thermodynamics. In the process we will also uncover a simple statistical interpretation of the entropy function in thermodynamics.
The grand canonical ensemble
In the previous chapter we considered a statistical system with a fixed number N of identical molecules. We have argued that although the energy E of the system is a constant its precise value is not known. Hence we considered the probability P(E) that the system had energy E and used it to relate the average value of the energy of the system (involving the microscopic properties of the system) to the macroscopic thermodynamic variable U, the internal energy. In this section we will generalize this approach to include a variable number of molecules, Figure 3.1. We note that the number of particles N in a volume, although a constant, is similarly not precisely known.