Statistical Mechanics - Statistical Ensembles

Statistical Ensembles

The modern formulation of statistical mechanics is based on the description of the physical system by an ensemble that represents all possible configurations of the system and the probability of realizing each configuration.

Each ensemble is associated with a partition function that, with mathematical manipulation, can be used to extract values of thermodynamic properties of the system. According to the relationship of the system to the rest of the universe, one of three general types of ensembles may apply, in order of increasing complexity:

• Microcanonical ensemble: describes a completely isolated system, having constant energy, as it does not exchange energy or mass with the rest of the universe.
• Canonical ensemble: describes a system in thermal equilibrium with its environment. It may only exchange energy in the form of heat with the outside.
• Grand-canonical ensemble: used in open systems which exchange energy and mass with the outside.

Summary of ensembles	Ensembles
	Microcanonical	Canonical	Grand canonical
Variables (suppressed constant for ensemble)	E, N, V	T, N, V	T, μ, V
Microscopic features	Number of microstates	Canonical partition function	Grand canonical partition function
Macroscopic function