Editing Equipartition theorem (section)

===General proofs===
General derivations of the equipartition theorem can be found in many [[statistical mechanics]] textbooks, both for the [[microcanonical ensemble]]<ref name="pathria_1972" /><ref name="huang_1987" /> and for the [[canonical ensemble]].<ref name="pathria_1972" /><ref name="tolman_1938" />
They involve taking averages over the [[phase space]] of the system, which is a [[symplectic manifold]].

To explain these derivations, the following notation is introduced. First, the phase space is described in terms of [[canonical coordinates|generalized position coordinates]] {{math|''q''<sub>''j''</sub>}} together with their [[conjugate momentum|conjugate momenta]] {{math|''p''<sub>''j''</sub>}}. The quantities {{math|''q''<sub>''j''</sub>}} completely describe the [[Configuration space (physics)|configuration]] of the system, while the quantities {{math|(''q''<sub>''j''</sub>,''p''<sub>''j''</sub>)}} together completely describe its [[Classical mechanics|state]].

Secondly, the infinitesimal volume
<math display="block">d\Gamma = \prod_i  dq_i \, dp_i \,</math>
of the phase space is introduced and used to define the volume {{math|Σ(''E'', Δ''E'')}} of the portion of phase space where the energy {{mvar|H}} of the system lies between two limits, {{mvar|E}} and {{math|''E'' + Δ''E''}}:
<math display="block">\Sigma (E, \Delta E) = \int_{H \in \left[E, E+\Delta E \right]} d\Gamma .</math>
In this expression, {{math|Δ''E''}} is assumed to be very small, {{math|Δ''E'' ≪ ''E''}}. Similarly, {{math|Ω(''E'')}} is defined to be the total volume of phase space where the energy is less than {{mvar|E}}:
<math display="block">\Omega (E) = \int_{H < E} d\Gamma.</math>

Since {{math|Δ''E''}} is very small, the following integrations are equivalent
<math display="block">\int_{H \in \left[ E, E+\Delta E \right]} \ldots d\Gamma  = \Delta E \frac{\partial}{\partial E} \int_{H < E} \ldots d\Gamma,</math>
where the ellipses represent the integrand. From this, it follows that {{math|Σ}} is proportional to {{math|Δ''E''}}
<math display="block">\Sigma(E, \Delta E) = \Delta E \ \frac{\partial \Omega}{\partial E} = \Delta E \ \rho(E),</math>
where {{math|''ρ''(''E'')}} is the [[density of states]]. By the usual definitions of [[statistical mechanics]], the [[entropy]] {{mvar|S}} equals {{math|''k''<sub>B</sub> log Ω(''E'')}}, and the [[temperature]] {{mvar|T}} is defined by
<math display="block">\frac{1}{T} = \frac{\partial S}{\partial E} = k_\text{B} \frac{\partial \log \Omega}{\partial E} = k_\text{B} \frac{1}{\Omega}\,\frac{\partial \Omega}{\partial E} .</math>

====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]].

====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.