The Grand Canonical Ensemble and Grand Partition Function

This chapter introduces the concept of the grand canonical ensemble using the example of the system of oscillators to develop the statistical thermodynamics of a system in contact with a reservoir that provides particles as well as energy. The ensemble of system microstates that emerges is designed to capture the statistical properties of a system coupled to a heat and particle bath. This is very suitable for a description of coexistence between phases such as a gas and a liquid. The principal role for the grand canonical ensemble is to enable us to understand how the reservoir chemical potential controls the mean number of particles in a system, and how that number might fluctuate. The main purpose of the grand partition function is that it allows ensemble averages to be obtained by differentiation. The chapter also finds that vacancy formation in crystals can be naturally treated with these methods.