Editing Equipartition theorem (section)

====The canonical ensemble====
In the [[canonical ensemble]], the system is in [[thermal equilibrium]] with an infinite heat bath at [[temperature]] {{mvar|T}} (in kelvins).<ref name="pathria_1972" /><ref name="tolman_1938" /> The probability of each state in [[phase space]] is given by its [[Boltzmann factor]] times a [[normalization factor]] <math>\mathcal{N}</math>, which is chosen so that the probabilities sum to one
<math display="block">\mathcal{N} \int e^{-\beta H(p, q)} d\Gamma = 1,</math>
where {{math|1=''β'' = 1/(''k''<sub>B</sub>''T'')}}. Using [[Integration by parts]] for a phase-space variable {{math|''x<sub>k</sub>''}} the above can be written as
<math display="block">
\mathcal{N} \int e^{-\beta H(p, q)} d\Gamma = \mathcal{N} \int d[x_k e^{-\beta H(p, q)}] d\Gamma_k - \mathcal{N} \int x_k \frac{\partial e^{-\beta H(p, q)}}{\partial x_k} d\Gamma,
</math>
where {{math|1=''d''Γ<sub>''k''</sub> = ''d''Γ/''dx<sub>k</sub>''}}, i.e., the first integration is not carried out over {{math|''x<sub>k</sub>''}}. Performing the first integral between two limits {{mvar|a}} and {{mvar|b}} and simplifying the second integral yields the equation
<math display="block">
\mathcal{N} \int  \left[ e^{-\beta H(p, q)} x_{k} \right]_{x_{k}=a}^{x_{k}=b} d\Gamma_{k}+ 
\mathcal{N} \int  e^{-\beta H(p, q)} x_{k} \beta \frac{\partial H}{\partial x_{k}} d\Gamma = 1,
</math>

The first term is usually zero, either because {{math|''x<sub>k</sub>''}} is zero at the limits, or because the energy goes to infinity at those limits. In that case, the equipartition theorem for the canonical ensemble follows immediately
<math display="block">
\mathcal{N} \int e^{-\beta H(p, q)} x_k \frac{\partial H}{\partial x_{k}} \,d\Gamma = 
\left\langle x_k \frac{\partial H}{\partial x_k} \right\rangle = \frac{1}{\beta} = k_\text{B} T.
</math>

Here, the averaging symbolized by <math>\langle \ldots \rangle</math> is the [[ensemble average]] taken over the [[canonical ensemble]].