The Microcanonical Ensemble

In this chapter we show that conservative character of the thermodynamic field, hence the existence of a thermodynamic potential, emerges from the very Hamiltonian nature of the equation of motion in generic many-body Hamiltonian systems provided a crucial assumption is made, namely the ergodic hypothesis. The microcanonical ensemble is accordingly introduced and its main mathematical properties discussed, along with a discussion of the meaning of the ergodic hypothesis, its validity and its necessity for establishing a link between mechanics and thermodynamics. We also introduce and discuss the notions of metric indecomposability and invariant measure. The chapter concludes with the application of microcanonical theory to ideal gases and with some historical remarks.