Editing Canonical ensemble (section)

{{Use American English|date=January 2019}}
{{Use mdy dates|date=January 2019}}
{{Short description|Ensemble of states at a constant temperature}}
{{Statistical mechanics|cTopic=Ensembles}}

In [[statistical mechanics]], a '''canonical ensemble''' is the [[statistical ensemble (mathematical physics)|statistical ensemble]] that represents the possible states of a mechanical system in [[thermal equilibrium]] with a [[heat bath]] at a fixed temperature.<ref name="gibbs">{{cite book |last=Gibbs |first=Josiah Willard |author-link=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York|title-link=Elementary Principles in Statistical Mechanics }}</ref> The system can exchange energy with the heat bath, so that the states of the system will differ in total energy.

The principal thermodynamic variable of the canonical ensemble, determining the [[probability distribution]] of states, is the [[absolute temperature]] (symbol: {{math|''T''}}). The ensemble typically  also depends on mechanical variables such as the number of particles in the system (symbol: {{math|''N''}}) and the system's volume (symbol: {{math|''V''}}), each of which influence the nature of the system's internal states. An ensemble with these three parameters, which are assumed constant for the ensemble to be considered canonical, is sometimes called the '''{{math|''NVT''}} ensemble'''.

The canonical ensemble assigns a probability {{math|''P''}} to each distinct [[microstate (statistical mechanics)|microstate]] given by the following exponential:

:<math>P = e^{(F - E)/(k T)},</math>

where {{math|''E''}} is the total energy of the microstate, and {{math|''k''}} is the [[Boltzmann constant]].

The number {{math|''F''}} is the free energy (specifically, the [[Helmholtz free energy]]) and is assumed to be a constant for a specific ensemble to be considered canonical. However, the probabilities and {{math|''F''}} will vary if different ''N'', ''V'', ''T'' are selected. The free energy {{math|''F''}} serves two roles: first, it provides a normalization factor for the [[probability distribution]] (the probabilities, over the complete set of microstates, must add up to one); second, many important ensemble averages can be directly calculated from the function {{math|''F''(''N'', ''V'', ''T'')}}.

An alternative but equivalent formulation for the same concept writes the probability as

:<math>\textstyle P = \frac{1}{Z} e^{-E/(k T)},</math>

using the [[Partition function (statistical mechanics)|canonical partition function]]

:<math>\textstyle Z = e^{-F/(k T)}</math>

rather than the free energy. The equations below (in terms of free energy) may be restated in terms of the canonical partition function by simple mathematical manipulations.

Historically, the canonical ensemble was first described by [[Ludwig Boltzmann|Boltzmann]] (who called it a ''holode'') in 1884 in a relatively unknown paper.<ref>{{cite book | last = Cercignani | first = Carlo | author-link = Carlo Cercignani | title = Ludwig Boltzmann: The Man Who Trusted Atoms | publisher = Oxford University Press | year = 1998 | isbn = 9780198501541 | url-access = registration | url = https://archive.org/details/ludwigboltzmannm0000cerc }}</ref> It was later reformulated and extensively investigated by [[Josiah Willard Gibbs|Gibbs]] in 1902.<ref name="gibbs"/>