Chapter 4 reviews the basic notions of equilibrium statistical mechanics and begins with its fundamental postulate and outlines the structure of the theory using the most important of the various statistical ensembles, the microcanonical, the canonical, and the grand canonical ensemble and their corresponding thermodynamic potential, the internal energy, the Helmholtz free energy, and the grand canonical potential. The notions of temperature, pressure, and chemical potential are obtained and it introduces the laws of thermodynamics, the Gibbs entropy, and the concept of the partition function. It also discusses quantum statistical mechanics using the density matrix and as applied to non-interacting Bosonic and Fermionic quantum gases, the former showing Bose–Einstein condensation. The mean-field theory of interacting magnetic moments and the transfer matrix to exactly solve the Ising model in one dimension serve as applications.
Hierarchical mean-field theory in quantum statistical mechanics: A bosonic example