The Canonical Ensemble

In this chapter we discuss the canonical ensemble. We first show that it emerges as the ensemble that describes the statistics of a system in weak interaction with a large ideal gas bath. We show that, like the microcanonical ensemble, the canonical ensemble accounts for the conservativeness of the thermodynamic field and the existence of the thermodynamic potential, that is it provides a good mechanical model of thermodynamics. We derive the ideal gas thermodynamics within the canonical formalism, as well as general expressions linking correlation functions to response coefficients (i.e., the fluctuation-dissipation relations). We conclude with a discussion of the finite heat bath statistics, and the peculiar thermodynamic properties of logarithmic oscillators.