Canonical Ensemble

The canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system. For a system taking only discrete values of energy, the probability distribution is characterized by the probability of finding the system in a particular microscopic state with energy level, conditioned on the prior knowledge that the total energy of the system and reservoir combined remains constant. This is given by the Boltzmann distribution,

where

is the normalizing constant explained below (A is the Helmholtz free energy function). The Boltzmann distribution describes a system that can exchange energy with a heat bath (or alternatively with a large number of similar systems) so that its temperature remains constant. Equivalently, it is the distribution which has maximum entropy for a given average energy .

It is also referred to as the NVT ensemble: the number of particles and the volume of each system in the ensemble are constant and the ensemble has a well-defined temperature, given by the temperature of the heat bath with which it would be in equilibrium.

The quantity is the Boltzmann constant, which relates the units of temperature to units of energy. It may be suppressed by expressing the absolute temperature using thermodynamic beta,

.

The quantities and are constants for a particular ensemble, which ensure that is normalised to 1. is therefore given by

.

This is called the partition function of the canonical ensemble. Specifying this dependence of on the energies conveys the same mathematical information as specifying the form of above.

The canonical ensemble (and its partition function) is widely used as a tool to calculate thermodynamic quantities of a system under a fixed temperature. This article derives some basic elements of the canonical ensemble. Other related thermodynamic formulas are given in the partition function article. When viewed in a more general setting, the canonical ensemble is known as the Gibbs measure, where, because it has the Markov property of statistical independence, it occurs in many settings outside of the field of physics.