Editing Canonical ensemble (section)

== 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
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| 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.}}
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| image1    = Ensemble quantum 1DOF all states.png
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| 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>}}.
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| image2    = Ensemble quantum 1DOF canonical.png
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| caption2  = A canonical ensemble for this system, for the temperature shown. The states are weighted exponentially in energy.
}}

{{details|topic=the representation of ensembles in quantum mechanics|Statistical ensemble (mathematical physics)}}

A statistical ensemble in quantum mechanics is represented by a [[density matrix]], denoted by <math>\hat \rho</math>. In basis-free notation, the canonical ensemble is the density matrix{{citation needed|date=October 2013}}
: <math>\hat \rho = \exp\left(\tfrac{1}{kT}(F - \hat H)\right),</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|''F''}} is determined by the probability normalization condition that the density matrix has a [[trace (linear algebra)|trace]] of one, <math>\operatorname{Tr} \hat \rho=1</math>:
: <math>e^{-\frac{F}{k T}} = \operatorname{Tr} \exp\left(-\tfrac{1}{kT} \hat H\right).</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{F - E_i}{k T}} |\psi_i\rangle \langle \psi_i | </math>
: <math>e^{-\frac{F}{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
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| 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|topic=the representation of ensembles in classical mechanics|Statistical ensemble (mathematical physics)}}

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 monoatomic gas (not molecules), {{math|''n'' {{=}} 3''N''}}. 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{F - 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 the [[Planck 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|''F''}} provides a normalizing factor and is also the characteristic state function, the free energy.

Again, the value of {{math|''F''}} is determined by demanding that {{math|''ρ''}} is a normalized probability density function:
: <math>e^{-\frac{F}{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.