ABSTRACT
Thermodynamic relations in each of the ensembles are most naturally governed by the thermodynamic potential having the same variables that define the ensemble. This chapter aims to find the Boltzmann-Gibbs distribution, the equilibrium probability density function for an ensemble of closed systems in thermal contact with their surroundings and develop the ensemble formalism of statistical mechanics. Prior experience with quantum mechanics indicates that we should seek an operator to take the place of the phase-space probability functions of classical statistical mechanics. Statistical mechanics requires modification when the quantum nature of matter is taken into account. A requirement on statistical mechanics is that it reproduce the laws of thermodynamics, a demand ensured by equating macroscopically measurable quantities with appropriate ensemble averages. The grand canonical ensemble is a collection of identically prepared systems that allow not only the exchange of energy with the surroundings, but particles as well.
