A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. Equivalently, the members of the ensemble can be considered loosely-coupled to each other so that they can share the total energy. The distribution of the total energy amongst the possible dynamical states (i.e. the members of the ensemble) is given by the partition function. A generalization of this is the grand canonical ensemble, in which the systems may share particles as well as energy. By contrast, in the microcanonical ensemble, the energy of each individual system is fixed. Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... The microcanonical ensemble is the simplest of the ensembles of statistical mechanics, in which many replicas of a system are assumed to be confined to a region of phase space of constant energy. ... In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. ... The isothermal-isobaric ensemble is an statistical mechanical ensemble where the partition function is given as The characteristic state function of this ensemble is the Gibbs free energy since Categories: | ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In physics, a statistical ensemble is a very large set of similar systems, considered all at once. ... In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ... In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. ... The microcanonical ensemble is the simplest of the ensembles of statistical mechanics, in which many replicas of a system are assumed to be confined to a region of phase space of constant energy. ...

A full development of the concept of the canonical ensemble is given in the article on the partition function. An example of the mathematical formulation of the canonical ensemble as a probability measure expressed in the language of measure theory is given in the article on the Potts model. In number theory, see Partition function (number theory) In statistical mechanics, see Partition function (statistical mechanics) In quantum field theory, see Partition function (quantum field theory) In game theory, see Partition function (game theory) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share... In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ... In mathematics, a measure is a function that assigns a number, e. ... In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. ...

Physical considerations

The canonical ensemble allows one to determine quantitatively the behaviour of two interacting systems - provided that one is small when compared to the other. The word "interacting" in this context means that the two systems are not isolated from each other.

Derivation

Define the following: Image File history File links CanonicalEnsemble. ...

• S - the system of interest
• S' - the heat resevoir in which S resides; S is small compared to S'
• S* - the system consisting of S and S' combined together
• m - an indexing variable which labels all the available energy states of the system
• E_m - the energy of the state corresponding to the index m for the system S
• E' - the energy associated with the heat bath
• E* - the energy associated with S*
• Ω'(.) - a continuous function which denotes an estimate of the number of states available at a particular energy for the heat resevoir. For example, Ω'(E) denotes the number of states in the system between the energies E and E + δE, where δE is sufficiently small compared to the separation of energy levels, but also sufficiently large so that the energy interval contains a countable number of energy states.

Once S is placed in contact with S', a certain amount of time will pass after which the system is effectively in equilibrium. The resulting canonical ensemble allows one to calculate the set of probabilities p_m that S is in a particular energy state E_m.

From these definitions, the total energy of the system S* is given by

E* = E' + E_m

Suppose S is in a single state indexed by m, such that the ratio of the total number of accessible states in S* and S' is a constant denoted by C'. Then

p_m = C'Ω'(E')

Hence,

lnp_m = lnC' + lnΩ'(E') = lnC' + lnΩ'(E * − E_m)

Since E_m is small compared to E*, a Taylor series expansion can be performed on the latter logarithm around the the energy E'. An appropriate approximation can be obtained by keeping the first two terms of the Taylor series expansion: As the degree of the Taylor series rises, it approaches the correct function. ... As the degree of the Taylor series rises, it approaches the correct function. ...

The following quantity is a constant which is traditionally denoted by β

Finally,

lnp_m = lnC' + lnΩ'(E ^* ) − βE_m.

Exponentiating this expression gives

The factor in front of the exponential can be treated as a normalization constant C, where

C = C'Ω'(E ^* ).

From this

Normalization to recover the partition function

Since probabilities must sum to unity, it must be the case that The word unity simply means oneness and is used in a variety of ways. ...

where by definition

is known as the partition function for the canonical ensemble. In number theory, see Partition function (number theory) In statistical mechanics, see Partition function (statistical mechanics) In quantum field theory, see Partition function (quantum field theory) In game theory, see Partition function (game theory) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share...

Characteristic state function

The characteristic state function of the canonical ensemble is the Helmholtz free energy function, as the following relationship holds: This page develops the Helmholtz free energy from the point of view of thermal and statistical physics. ...