This chapter is concerned with Gibbs’ statistical mechanics. It relies on developing the constraints imposed by Hamiltonian mechanics on the time evolution of a general probability density function in phase space. This is effectively done by using the notion of Hamiltonian flow and material derivative. Combining conservation of probability with Liouville’s theorem of Hamiltonian mechanics gives rise to Liouville’s equation, which is a cornerstone equation of both time-dependent and equilibrium statistical mechanics. From there on, the chapter focuses on equilibrium statistical mechanics and introduces the canonical and microcanonical Gibbs’ ensembles. The chapter takes a step-by-step approach where the main ideas are presented first for one particle in one dimension of space, and then reformulated in more increasingly more complex situations. Important properties such as the partition function acting as a moment generating function are derived and put in practice. Finally, a whole section is dedicated to little know works from Gibbs on statistical mechanics for identical particles. Finally, the grand canonical ensemble is also introduced.

Statistical ensembles provide a conceptual framework within whichto obtain the average behaviour of physical systems.The extent to which a system interacts with its environment determines the appropriate statistical ensemble with which to describe its properties.Isolated systems are described by the microcanonical ensemble.The expression for the Boltzmann entropy acts as a bridge equation relating the thermodynamic quantity, the entropy, to a statistical mechanical quantity, the multiplicity of microstates.Other forms of entropy, the Gibbs and Shannon entropies, and the relation of entropy to irreversibility are discussed.