Introduction and definitions

Probability theory treats the properties of a completely abstract idealization called an ensemble. An ensemble is a collection of members, each of which has certain properties, and they obey a set of external parameters. Depending on the problem, the ensemble may contain just a few members, many members, or an infinite number of members.

In statistical mechanics one computes the probabilities of the various possible results of a variety of measurements made on a system. Also from the viewpoint of statistical mechanics, a system is in condition of equilibrium when the information one has about it has reached a time-independent minimum.The information is usually one of the following three sets:

1. The system consists of N particles of known type or types contained in a volume V. It is completely isolated from the rest of the world. This means that its energy was constant from the moment it has reached equilibrium. The isolated equilibrium system is therefore completely described by N,V,and E. The ensemble used to represent this information is called microcanonical.

2. The system consists of N particles of known type or types contained in a volume V. It is in thermal contact with a heat reservoir, which is simply a huge equilibrium system at temperature T. This means that although energy fluctuations are possible for the system, since it can exchange energy with the heat reservoir, its temperature at equilibrium will by definition be that of the reservoir. The equilibrium system in thermal contact with a heat reservoir is therefore completely described by N,V,and T. The ensemble used to represent this information is called canonical.

3. The system consists of the material contained in a volume V. This material is in thermal contact with a large heat reservoir characterized by the temperature T. The system is also open to the exchange of each type of matter with a particle reservoir or bath for each type. These are simply huge equilibrium systems containing paticles with chemical potential $\mu$ for each type, and in contact with the system through a permeable wall.The equilibrium system in thermal contact with a heat reservoir and open to the exchange of matter with a particle bath is therefore completely describeed by $\mu$,V,and T. The ensemble used to represent this information is called grand canonical [3].