Equilibrium Ensembles

The main tools of Statistical Mechanics are the equilibrium ensembles. In this chapter we introduce and study the standard ensembles, their partition functions, associated thermodynamical potentials, probability distributions, and thermodynamic limits. We discuss how the partition functions of the diverse ensembles are related by a web of Laplace transforms, and how these transforms reduce in the thermodynamic limit to Legendre transforms between the corresponding potentials as required by Thermodynamics. We start by introducing the microcanonical ensemble for either classical or quantum system, discrete and continuous. This leads us to Boltzmann’s statistical interpretation of entropy and the related results. We then digress on the modern interpretations of entropy, introducing the information-theoretic Shannon entropy and the quantum von Neumann entropy: these more conceptual viewpoints shed light on the logic foundations of Statistical Mechanics. We then introduce the canonical ensemble and study it in detail in the classical and in the quantum set-ups: we prove the Bohr–Van Leeuwen, the virial and equipartition theorems, and study the classical heat capacities. We introduce the pressure ensemble, and discuss the Feynman gas as an application of it. Finally we define the Grand canonical ensemble and study its main properties