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{{Statistical mechanics|cTopic=[[Statistical ensemble (mathematical physics)|Ensembles]]}}

In [[statistical mechanics]], a '''canonical ensemble''' is the [[statistical ensemble (mathematical physics)|statistical ensemble]] that is used to represent the possible states of a mechanical system which is in [[thermal equilibrium]] with a heat bath.<ref name="gibbs">{{cite book |last=Gibbs |first=Josiah Willard |authorlink=Josiah Willard Gibbs |title=[[Elementary Principles in Statistical Mechanics]] |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York}}</ref> The system is said to be closed in the sense that the system can exchange energy with a heat bath, so that various possible states of the system can differ in total energy. The system's composition, volume, and shape are kept the same in all possible states of the system.

The thermodynamic variable of the canonical ensemble is the [[absolute temperature]] (symbol: {{math|''T''}}). The ensemble is also dependent on mechanical variables such as the number of particles in the system (symbol: {{math|''N''}}) and the system's volume (symbol: {{math|''V''}}), each which influence the nature of the system's internal states. This ensemble is therefore sometimes called the '''{{math|''NVT''}} ensemble''', as each of these three quantities is a constant of the ensemble.

In simple terms, the canonical ensemble assigns a probability {{math|''P''}} to each distinct [[microstate (statistical mechanics)|microstate]] given by the following exponential:
:<math>P = e^{\frac{A - E}{k T}},</math>
where {{math|''E''}} is the total energy of the microstate, and {{math|''k''}} is [[Boltzmann's constant]].

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

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 isbn|9780198501541}}</ref> It was later reformulated and extensively investigated by [[Josiah Willard Gibbs|Gibbs]] in 1902.<ref name="gibbs"/> 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^{-A/(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.

==Applicability of canonical ensemble==

The canonical ensemble is exactly the ensemble that describes the possible states of an isolated system that is in thermal equilibrium with a heat bath (the derivation of this fact can be found in Gibbs<ref name="gibbs"/>).
In this case the canonical ensemble applies exactly to systems of any size; while it is necessary to assume that the heat bath is very large (i. e., in the [[macroscopic limit]]), the system itself may be small or large.

The condition that the system is isolated is necessary in order to ensure it has well-defined equation of motion.<ref name="gibbs"/> In practice, however, it is desirable to apply the canonical ensemble to describe systems that are in direct contact with the heat bath, since it is that contact that ensures the equilibrium. The use of the canonical ensemble in these cases is usually justified either 1) by assuming that the contact is weak, or 2) by incorporating a part of the heat bath connection into the system under analysis, so that the connection's influence on the region of interest is correctly modelled.

When the total energy is exactly known but the internal state of the system is otherwise completely unknown, a better description is the [[microcanonical ensemble]]. Conversely, for equilibrium systems where the particle number is not fixed (due to a connection to a particle reservoir), a better description is found in the [[grand canonical ensemble]]. For large systems (in the [[thermodynamic limit]]) these other ensembles become essentially equivalent to the canonical ensemble, at least for average quantities.

==Properties==

{{unordered list
| ''Uniqueness'': The canonical ensemble is uniquely determined for a given system at a given temperature, and does not depend on arbitrary choices such as choice of coordinate system (classical mechanics), of basis (quantum mechanics), or of the zero of energy.<ref name="gibbs"/>
| ''Statistical equilibrium'' (steady state): A canonical ensemble does not evolve over time, despite the fact that the underlying system is in constant motion. This is because the ensemble is only a function of a conserved quantity of the system (energy).<ref name="gibbs"/>
| ''Thermal equilibrium with other systems'': Two systems, each described by a canonical ensemble of equal temperature, brought into thermal contact<ref group=note>Thermal contact means that the systems are made able to exchange energy through an interaction. The interaction must be weak as to not significantly disturb the systems' microstates.</ref> will each retain the same ensemble and the resulting combined system is described by a canonical ensemble of the same temperature.<ref name="gibbs"/>
|''Maximum entropy'': For a given mechanical system (fixed {{math|''N''}}, {{math|''V''}}), the canonical ensemble average {{math|−⟨log ''P''⟩}} is the maximum possible of any ensemble with the same {{math|⟨''E''⟩}}.<ref name="gibbs"/>
| ''Minimum free energy'': For a given mechanical system (fixed {{math|''N''}}, {{math|''V''}}) and given value of {{math|''T''}}, the canonical ensemble average {{math|⟨''E'' + ''kT'' log ''P''⟩}} is the lowest possible of any ensemble.<ref name="gibbs"/>
| The partial derivatives of the function {{math|''A''(''N'', ''V'', ''T'')}} give important canonical ensemble average quantities:
{{unordered list
|the average pressure is<ref name="gibbs"/>
:<math> \langle p \rangle = -\frac{\partial A} {\partial V}, </math>
|the [[Gibbs entropy]] is<ref name="gibbs"/>
:<math> S = -k \langle \log P \rangle = - \frac{\partial A} {\partial T}, </math>
|the partial derivative {{math|∂''A''/∂''N''}} is approximately related to [[chemical potential]], although the concept of chemical equilibrium does not exactly apply to canonical ensembles of small systems.<ref group=note>Since {{math|''N''}} is an integer, this "derivative" actually refers to a [[finite difference]] expression such as {{math|''A''(''N'') − ''A''(''N'' − 1)}}, or {{math|''A''(''N'' + 1) − ''A''(''N'')}}, or {{math|[''A''(''N'' + 1) − ''A''(''N'' − 1)]/2}}. These finite difference expressions are equivalent only in the thermodynamic limit (very large {{math|''N''}}).</ref>
|and the average energy is<ref name="gibbs"/>
:<math> \langle E \rangle = A + ST.</math>
}}
| ''Exact differential'': From the above expressions, it can be seen that the function {{math|''A''(''V'', ''T'')}}, for a given {{math|''N''}}, has the [[exact differential]]<ref name="gibbs"/>
:<math> dA = - S dT - \langle p\rangle dV .</math>
| ''First law of thermodynamics'': Substituting the above relationship for for {{math|⟨''E''⟩}} into the exact differential of {{math|''A''}}, an equation similar to the [[first law of thermodynamics]] is found, except with average signs on some of the quantities:<ref name="gibbs"/>
:<math> d\langle E \rangle = T dS - \langle p\rangle dV .</math>
| ''[[Thermal fluctuations|Energy fluctuations]]'': The energy in the system has uncertainty in the canonical ensemble. The [[variance]] of the energy is<ref name="gibbs"/>
:<math> \langle E^2 \rangle - \langle E \rangle^2 = k T^2 \frac{\partial \langle E \rangle}{\partial T}.</math>
}}

==Example ensembles==

=== Boltzmann distribution (separable system) ===

If a system described by a canonical ensemble can be separated into independent parts (this happens if the different parts do not interact), and each of those parts has a fixed material composition, then each part can be seen as a system unto itself and is described by a canonical ensemble having the same temperature as the whole. Moreover, if the system is made up of multiple ''similar'' parts, then each part has exactly the same distribution as the other parts.

In this way, the canonical ensemble provides exactly the [[Boltzmann distribution]] (also known as [[Maxwell–Boltzmann statistics]]) for systems of ''any number'' of particles. In comparison, the justification of the Boltzmann distribution from the [[microcanonical ensemble]] only applies for systems with a large number of parts (that is, in the therymodynamic limit).

The Boltzmann distribution itself is one of the most important tools in applying statistical mechanics to real systems, as it massively simplifies the study of systems that can be separated into independent parts (e. g., [[Maxwell speed distribution|particles in a gas]], [[Planck's law|electromagnetic modes in a cavity]], [[polymer physics|molecular bonds in a polymer]]).

=== Ising model (strongly interacting system) ===

{{main|Ising model}}

In a system composed of pieces that interact with each other, it is usually not possible to find a way to separate the system into independent subsystems as done in the Boltzmann distribution. In these systems it is necessary to resort to using the full expression of the canonical ensemble in order to describe the thermodynamics of the system when it is thermostatted to a heat bath. The canonical ensemble is generally the most straightforward framework for studies of statistical mechanics and even allows one to obtain exact solutions in some interacting model systems.<ref>{{cite isbn|9780120831807}}</ref>

A classic example of this is the [[Ising model]], which is a widely discussed toy model for the phenomena of [[ferromagnetism]] and of [[self-assembled monolayer]] formation, and is one of the simplest models that shows a [[phase transition]]. [[Lars Onsager]] famously calculated exactly the free energy of an infinite-sized [[square-lattice Ising model]] at zero magnetic field, in the canonical ensemble.<ref>{{cite doi|10.1103/PhysRev.65.117}}</ref>

{{Expand section|1=another example of non-separable system besides Ising|date=November 2013}}

==Precise expressions for the ensemble==

The precise mathematical expression for a statistical ensemble depends on the kind of mechanics under consideration—quantum or classical—since the notion of a "microstate" is considerably different in these two cases. In quantum mechanics, the canonical ensemble affords a simple description since [[Matrix diagonalization|diagonalization]] provides a discrete set of [[microstate (statistical mechanics)|microstate]]s with specific energies. The classical mechanical case is more complex as it involves instead an integral over canonical [[phase space]], and the size of microstates in phase space can be chosen somewhat arbitrarily.

=== Quantum mechanical ===

{{multiple image
<!-- Essential parameters -->
| align     = right
| direction = horizontal
| width     = 220
| header    = Example of canonical ensemble for a quantum system consisting of one particle in a potential well.
| footer    = {{small|The particle's Hamiltonian is [[Schrödinger equation|Schrödinger]]-type, {{math|''Ĥ'' {{=}} ''U''(''x'') + ''p''<sup>2</sup>/2''m''}} (the potential {{math|''U''(''x'')}} is plotted as a red curve). Each panel shows an energy-position plot with the various stationary states, along with a side plot showing the distribution of states in energy.}}
<!-- Image 1 -->
| image1    = Ensemble quantum 1DOF all states.png
| width1    = 
| alt1      = 
| caption1  = Plot of all possible states of this system. The available stationary states displayed as horizontal bars of varying darkness according to {{math|{{!}}''ψ''<sub>''i''</sub>(x){{!}}<sup>2</sup>}}.
<!-- Image 2 -->
| image2    = Ensemble quantum 1DOF canonical.png
| width2    = 
| alt2      = 
| caption2  = A canonical ensemble for this system, for the temperature shown. The states are weighted exponentially in energy.
}}

{{Details|statistical ensemble (mathematical physics)|the representation of ensembles in quantum mechanics}}

A statistical ensemble in quantum mechanics is represented by a [[density matrix]], denoted by {{math|''ρ̂''}}. In basis-free notation, the canonical ensemble is the density matrix{{citation needed|date=October 2013}}
:<math>\hat \rho = \exp\big(\tfrac{1}{kT}(A - \hat H)\big),</math>
where {{math|''Ĥ''}} is the system's total energy operator ([[Hamiltonian (quantum mechanics)|Hamiltonian]]), and {{math|exp()}} is the [[matrix exponential]] operator. The free energy {{math|''A''}} is determined by the probability normalization condition that the density matrix has a [[trace (linear algebra)|trace]] of one, {{math|Tr ''ρ̂'' {{=}} 1}}:
:<math>e^{-\frac{A}{k T}} = \operatorname{Tr} \exp\big(-\tfrac{1}{kT} \hat H\big).</math>

The canonical ensemble can alternatively be written in a simple form using [[bra-ket notation]], if the system's [[stationary state|energy eigenstates]] and energy eigenvalues are known. Given a complete basis of energy eigenstates {{math|{{!}}''ψ''<sub>''i''</sub>⟩}}, indexed by {{math|''i''}}, the canonical ensemble is:
:<math>\hat \rho = \sum_i e^{\frac{A - E_i}{k T}} |\psi_i\rangle \langle \psi_i | </math>
:<math>e^{-\frac{A}{k T}} = \sum_i e^{\frac{- E_i}{k T}}.</math>
where the {{math|''E''<sub>''i''</sub>}} are the energy eigenvalues determined by {{math|''Ĥ''{{!}}''ψ''<sub>''i''</sub>⟩ {{=}} ''E''<sub>''i''</sub>{{!}}''ψ''<sub>''i''</sub>⟩}}. In other words, a set of microstates in quantum mechanics is given by a complete set of stationary states. The density matrix is diagonal in this basis, with the diagonal entries each directly giving a probability.

===Classical mechanical===

{{multiple image
<!-- Essential parameters -->
| align     = right
| direction = horizontal
| width     = 220
| header    = Example of canonical ensemble for a classical system consisting of one particle in a potential well.
| footer    = Each panel shows [[phase space]] (upper graph) and energy-position space (lower graph). The particle's Hamiltonian is {{math|''H'' {{=}} ''U''(''x'') + ''p''<sup>2</sup>/2''m''}}, with the potential {{math|''U''(''x'')}} shown as a red curve. The side plot shows the distribution of states in energy.
<!-- Image 1 -->
| image1    = Ensemble classical 1DOF all states.png
| width1    = 
| alt1      = 
| caption1  = Plot of all possible states of this system. The available physical states are evenly distributed in phase space, but with an uneven distribution in energy; the side-plot displays {{math|''dv''/''dE''}}.
<!-- Image 2 -->
| image2    = Ensemble classical 1DOF canonical.png
| width2    = 
| alt2      = 
| caption2  = A canonical ensemble for this system, for the temperature shown. The states are weighted exponentially in energy.
}}

{{Details|statistical ensemble (mathematical physics)|the representation of ensembles in classical mechanics}}

In classical mechanics, a statistical ensemble is instead represented by a [[joint probability density function]] in the system's [[phase space]], 
{{math|''ρ''(''p''<sub>1</sub>, … ''p''<sub>''n''</sub>, ''q''<sub>1</sub>, … ''q''<sub>''n''</sub>)}}, where the {{math|''p''<sub>1</sub>, … ''p''<sub>''n''</sub>}} and {{math|''q''<sub>1</sub>, … ''q''<sub>''n''</sub>}} are the [[canonical coordinates]] (generalized momenta and generalized coordinates) of the system's internal degrees of freedom.
In a system of particles, the number of degrees of freedom {{math|''n''}} depends on the number of particles {{math|''N''}} in a way that depends on the physical situation. For a three dimensional gas of monoatoms (not molecules), {{math|''n'' {{=}} 3''N''}}, however in diatomic gases there will also be rotational and vibrational degrees of freedom.

The probability density function for the canonical ensemble is:
:<math>\rho = \frac{1}{h^n C} e^{\frac{A - E}{k T}},</math>
where
* {{math|''E''}} is the energy of the system, a function of the phase {{math|(''p''<sub>1</sub>, … ''q''<sub>''n''</sub>)}},
* {{math|''h''}} is an arbitrary but predetermined constant with the units of {{math|energy×time}}, setting the extent of one microstate and providing correct dimensions to {{math|''ρ''}}.<ref group=note>(Historical note) Gibbs' original ensemble effectively set {{math|''h'' {{=}} 1 [energy unit]×[time unit]}}, leading to unit-dependence in the values of some thermodynamic quantities like entropy and chemical potential. Since the advent of quantum mechanics, {{math|''h''}} is often taken to be equal to [[Planck's constant]] in order to obtain a semiclassical correspondence with quantum mechanics.</ref>
* {{math|''C''}} is an overcounting correction factor, often used for particle systems where identical particles are able to change place with each other.<ref group=note>In a system of {{math|''N''}} identical particles, {{math|''C'' {{=}} ''N''!}} ([[factorial]] of {{math|''N''}}). This factor corrects the overcounting in phase space due to identical physical states being found in multiple locations. See the [[Statistical ensemble (mathematical physics)#Correcting overcounting in phase space|statistical ensemble]] article for more information on this overcounting.</ref>
* {{math|''A''}} provides a normalizing factor and is also the characteristic state function, the free energy.

Again, the value of {{math|''A''}} is determined by demanding that {{math|''ρ''}} is a normalized probability density function:
:<math>e^{-\frac{A}{k T}} = \int \ldots \int \frac{1}{h^n C} e^{\frac{- E}{k T}} \, dp_1 \ldots dq_n </math>
This integral is taken over the entire [[phase space]].

In other words, a microstate in classical mechanics is a phase space region, and this region has volume {{math|''h<sup>n</sup>C''}}. This means that each microstate spans a range of energy, however this range can be made arbitrarily narrow by choosing {{math|''h''}} to be very small. The phase space integral can be converted into a summation over microstates, once phase space has been finely divided to a sufficient degree.

== Notes ==
{{reflist|group=note}}

== References ==
{{reflist}}

{{Statistical mechanics topics}}

[[Category:Statistical ensembles]]