Editing Equipartition theorem (section)

===Ideal gas law===
{{See also|Ideal gas|Ideal gas law}}

[[Ideal gas]]es provide an important application of the equipartition theorem. As well as providing the formula
<math display="block">
\begin{align}
\langle H^{\mathrm{kin}} \rangle &= \frac{1}{2m} \langle p_{x}^{2} + p_{y}^{2} + p_{z}^{2} \rangle\\
&=  \frac{1}{2} \left(
\left\langle p_{x} \frac{\partial H^{\mathrm{kin}}}{\partial p_{x}} \right\rangle + 
\left\langle p_{y} \frac{\partial H^{\mathrm{kin}}}{\partial p_{y}} \right\rangle +
\left\langle p_{z} \frac{\partial H^{\mathrm{kin}}}{\partial p_{z}} \right\rangle \right) = 
\frac{3}{2} k_\text{B} T
\end{align}
</math>
for the average kinetic energy per particle, the equipartition theorem can be used to derive the [[ideal gas law]] from classical mechanics.<ref name="pathria_1972" /> If '''q''' = (''q<sub>x</sub>'', ''q<sub>y</sub>'', ''q<sub>z</sub>'') and '''p''' = (''p<sub>x</sub>'', ''p<sub>y</sub>'', ''p<sub>z</sub>'') denote the position vector and momentum of a particle in the gas, and
'''F''' is the net force on that particle, then
<math display="block">
\begin{align}
\langle \mathbf{q} \cdot \mathbf{F} \rangle &= \left\langle q_x \frac{dp_x}{dt} \right\rangle + 
\left\langle q_y \frac{dp_y}{dt} \right\rangle + 
\left\langle q_z \frac{dp_z}{dt} \right\rangle\\
&=-\left\langle q_x \frac{\partial H}{\partial q_x} \right\rangle -
\left\langle q_y \frac{\partial H}{\partial q_y} \right\rangle - 
\left\langle q_z \frac{\partial H}{\partial q_z} \right\rangle = -3k_\text{B} T,
\end{align}
</math>
where the first equality is [[Newton's second law]], and the second line uses [[Hamilton's equations]] and the equipartition formula. Summing over a system of ''N'' particles yields
<math display="block">
3Nk_\text{B} T = - \left\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \right\rangle.
</math>
[[Image:Translational motion.gif|frame|right|Figure 5. The kinetic energy of a particular molecule can [[thermal fluctuations|fluctuate wildly]], but the equipartition theorem allows its ''average'' energy to be calculated at any temperature. Equipartition also provides a derivation of the [[ideal gas law]], an equation that relates the [[pressure]], [[volume]] and [[temperature]] of the gas. (In this diagram five of the molecules have been colored red to track their motion; this coloration has no other significance.)]]

By [[Newton's third law]] and the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure ''P'' of the gas. Hence
<math display="block">
-\left\langle\sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k}\right\rangle = P \oint_{\text{surface}} \mathbf{q} \cdot d\mathbf{S},
</math>
where {{math|''d'''''S'''}} is the infinitesimal area element along the walls of the container. Since the [[divergence]] of the position vector {{math|'''q'''}} is
<math display="block">
\boldsymbol\nabla \cdot \mathbf{q} =
\frac{\partial q_x}{\partial q_x} + 
\frac{\partial q_y}{\partial q_y} + 
\frac{\partial q_z}{\partial q_z} = 3,
</math>
the [[divergence theorem]] implies that
<math display="block">P \oint_{\mathrm{surface}} \mathbf{q} \cdot \mathbf{dS} = P \int_{\mathrm{volume}} \left( \boldsymbol\nabla \cdot \mathbf{q} \right) \, dV = 3PV,</math>
where {{math|''dV''}} is an infinitesimal volume within the container and {{mvar|V}} is the total volume of the container.

Putting these equalities together yields
<math display="block">3Nk_\text{B} T = -\left\langle \sum_{k=1}^N \mathbf{q}_k \cdot \mathbf{F}_k \right\rangle = 3PV,</math>
which immediately implies the [[ideal gas law]] for ''N'' particles:
<math display="block">PV = N k_\text{B} T = nRT,</math>
where {{math|1=''n'' = ''N''/''N''<sub>A</sub>}} is the number of moles of gas and {{math|1=''R'' = ''N''<sub>A</sub>''k''<sub>B</sub>}} is the [[gas constant]]. Although equipartition provides a simple derivation of the ideal-gas law and the internal energy, the same results can be obtained by an alternative method using the [[partition function (statistical mechanics)|partition function]].<ref name="configint">
Vu-Quoc, L., [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)], 2008. this wiki site is down; see [https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 this article in the web archive on 2012 April 28].
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L. Vu-Quoc, [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)], 2008.
--->
</ref>