The Boltzmann Factor and the Canonical Partition Function

The canonical ensemble is designed to represent the statistics of a system that is open to energy exchange with a large environment. It is mathematically easier to employ in practice than the microcanonical ensemble designed for isolated systems. This chapter illustrates some examples of its use, where it also encounters the important concept of the canonical partition function that plays a role in making the connection between classical and statistical thermodynamics. The Boltzmann factor plays a role in the statistical weighting of such a state in a canonical ensemble. One of the simplest systems studied in a canonical ensemble is a two-level paramagnet. The chapter considers the canonical statistical properties of a 1-d quantum harmonic oscillator using the partition function. The classical and quantum limits of the canonical statistical behaviour of an oscillator can be observed in the temperature dependence of the heat capacity of a diatomic molecular gas.