1.4 Statistical Ensembles
17
Fig. 1.5 (continued)
• Postulate: The long-time average of a mechanical variable in a thermodynamic
system is equal to its ensemble average in the limit that the number of members
of the ensemble tends to infinity.
Of course, before we can make use of this postulate, we shall need to be able to
compute an ensemble average. To accomplish this task, we will require a knowledge
of the relative probabilities of occurrence of the various (quantum) states making
up the individual members of the ensemble. We shall return to this topic in the
following chapter.
Many different ensembles may be constructed. However, because the three most
important thermodynamic environments can be described either as isolated systems
(with a fixed number of particles, N; fixed volume, V ; and fixed total energy, E)
or as a closed isothermal system (with fixed N, fixed V , and a fixed temperature,
T ) or as an open isothermal system (with fixed chemical potential, μ, fixed V , and
fixed T ), we shall focus mainly upon three representative ensembles, appropriately
referred to as the microcanonical ensemble (fixed N, V , E), the canonical ensemble
(fixed N, V , T ), and the grand ensemble (fixed μ, V , T ). In the event that we are
17
Fig. 1.5 (continued)
• Postulate: The long-time average of a mechanical variable in a thermodynamic
system is equal to its ensemble average in the limit that the number of members
of the ensemble tends to infinity.
Of course, before we can make use of this postulate, we shall need to be able to
compute an ensemble average. To accomplish this task, we will require a knowledge
of the relative probabilities of occurrence of the various (quantum) states making
up the individual members of the ensemble. We shall return to this topic in the
following chapter.
Many different ensembles may be constructed. However, because the three most
important thermodynamic environments can be described either as isolated systems
(with a fixed number of particles, N; fixed volume, V ; and fixed total energy, E)
or as a closed isothermal system (with fixed N, fixed V , and a fixed temperature,
T ) or as an open isothermal system (with fixed chemical potential, μ, fixed V , and
fixed T ), we shall focus mainly upon three representative ensembles, appropriately
referred to as the microcanonical ensemble (fixed N, V , E), the canonical ensemble
(fixed N, V , T ), and the grand ensemble (fixed μ, V , T ). In the event that we are