Editing Equipartition theorem (section)

====The microcanonical ensemble====
In the microcanonical ensemble, the system is isolated from the rest of the world, or at least very weakly coupled to it.<ref name="huang_1987" /> Hence, its total energy is effectively constant; to be definite, we say that the total energy {{mvar|H}} is confined between {{mvar|E}} and {{math|''E''+''dE''}}. For a given energy {{math|''E''}} and spread {{math|''dE''}}, there is a region of [[phase space]] {{math|Σ}} in which the system has that energy, and the probability of each state in that region of [[phase space]] is equal, by the definition of the microcanonical ensemble. Given these definitions, the equipartition average of phase-space variables {{math|''x<sub>m</sub>''}} (which could be either {{math|''q<sub>k</sub>''}} or {{math|''p<sub>k</sub>''}}) and {{math|''x<sub>n</sub>''}} is given by
:<math display="block">\begin{align}
\left\langle x_{m} \frac{\partial H}{\partial x_{n}} \right \rangle &=
\frac{1}{\Sigma}   \, \int_{H \in \left[ E, E+\Delta E \right]}  x_{m} \frac{\partial H}{\partial x_{n}} \,d\Gamma\\
&=\frac{\Delta E}{\Sigma}\, \frac{\partial}{\partial E} \int_{H < E}  x_{m} \frac{\partial H}{\partial x_{n}} \,d\Gamma\\
&= \frac{1}{\rho} \,\frac{\partial}{\partial E} \int_{H < E}  x_{m} \frac{\partial \left( H - E \right)}{\partial x_{n}} \,d\Gamma,
\end{align}</math>
where the last equality follows because {{math|''E''}} is a constant that does not depend on {{math|''x<sub>n</sub>''}}. [[Integration by parts|Integrating by parts]] yields the relation
<math display="block">\begin{align}
\int_{H < E} x_{m} \frac{\partial ( H - E )}{\partial x_{n}} \,d\Gamma &= 
\int_{H < E} \frac{\partial}{\partial x_{n}} \bigl( x_m ( H - E ) \bigr) \,d\Gamma - 
\int_{H < E} \delta_{mn} ( H - E ) d\Gamma \\
&= \delta_{mn} \int_{H < E} ( E - H ) \,d\Gamma,
\end{align}</math>
since the first term on the right hand side of the first line is zero (it can be rewritten as an integral of ''H'' − ''E'' on the [[hypersurface]] where {{math|1=''H'' = ''E''}}).

Substitution of this result into the previous equation yields
<math display="block">
\left\langle x_m \frac{\partial H}{\partial x_{n}} \right\rangle
= \delta_{mn} \frac{1}{\rho} \, \frac{\partial}{\partial E} \int_{H < E} \left( E - H \right)\,d\Gamma
= \delta_{mn} \frac{1}{\rho} \, \int_{H < E} \,d\Gamma
= \delta_{mn} \frac{\Omega}{\rho}.
</math>

Since <math> \rho = \frac{\partial \Omega}{\partial E} </math> the equipartition theorem follows:
<math display="block">
\left\langle x_{m} \frac{\partial H}{\partial x_{n}} \right\rangle = 
\delta_{mn} \left(\frac{1}{\Omega} \frac{\partial \Omega}{\partial E}\right)^{-1}  = 
\delta_{mn} \left(\frac{\partial \log \Omega} {\partial E}\right)^{-1} = \delta_{mn} k_\text{B} T.
</math>

Thus, we have derived the general formulation of the equipartition theorem
<math display="block">
\left\langle x_{m} \frac{\partial H}{\partial x_{n}} \right\rangle = \delta_{mn} k_\text{B} T,
</math>
which was so useful in the [[#Applications|applications]] described above.