Editing Equipartition theorem (section)

==Derivations==

===Kinetic energies and the Maxwell–Boltzmann distribution===
The original formulation of the equipartition theorem states that, in any physical system in [[thermal equilibrium]], every particle has exactly the same average translational [[kinetic energy]], {{math|{{sfrac|3|2}}''k''<sub>B</sub>''T''}}.<ref name="mcquarrie_2000a">{{cite book | last = McQuarrie | first = DA | year = 2000 | title = Statistical Mechanics | edition = revised 2nd | publisher = University Science Books | isbn = 978-1-891389-15-3 | pages = [https://archive.org/details/statisticalmecha00mcqu_0/page/121 121–128] | url = https://archive.org/details/statisticalmecha00mcqu_0/page/121 }}</ref> However, this is true only for [[ideal gas]], and the same result can be derived from the [[Maxwell–Boltzmann distribution]]. First, we choose to consider only the Maxwell–Boltzmann distribution of velocity of the z-component
<math display="block">f (v_z) = \sqrt{\dfrac{m}{2\pi k_\text{B}T}}\;e^{\frac{-m{v_z}^2}{2k_\text{B}T}}</math>

with this equation, we can calculate the mean square velocity of the {{mvar|z}}-component
<math display="block">\langle {v_z}^2 \rangle = \int_{-\infty}^{\infty} f (v_z){v_z}^2 dv_z = \dfrac{k_\text{B}T}{m}</math>

Since different components of velocity are independent of each other, the average translational kinetic energy is given by
<math display="block">\langle E_k \rangle = \dfrac 3 2 m \langle {v_z}^2 \rangle = \dfrac 3 2 k_\text{B}T</math>

Notice, the [[Maxwell–Boltzmann distribution]] should not be confused with the [[Boltzmann distribution]], which the former can be derived from the latter by assuming the energy of a particle is equal to its translational kinetic energy.

As stated by the equipartition theorem. The same result can also be obtained by averaging the particle energy using the probability of finding the particle in certain quantum energy state.<ref name="configint"/>

===Quadratic energies and the partition function===
More generally, the equipartition theorem states that any [[degrees of freedom (physics and chemistry)|degree of freedom]] {{mvar|x}} which appears in the total energy {{mvar|H}} only as a simple quadratic term {{math|''Ax''<sup>2</sup>}}, where {{mvar|A}} is a constant, has an average energy of {{math|{{1/2}}''k''<sub>B</sub>''T''}} in thermal equilibrium. In this case the equipartition theorem may be derived from the [[Partition function (statistical mechanics)|partition function]] {{math|''Z''(''β'')}}, where {{math|1=''β'' = 1/(''k''<sub>B</sub>''T'')}} is the canonical [[inverse temperature]].<ref>{{cite book | last = Callen | first = HB | author-link = Herbert Callen | year = 1985 | title = Thermodynamics and an Introduction to Thermostatistics | publisher = John Wiley and Sons | location = New York | pages = 375–377 | isbn = 0-471-86256-8}}</ref> Integration over the variable {{mvar|x}} yields a factor
<math display="block">Z_{x} = \int_{-\infty}^{\infty} dx \ e^{-\beta A x^{2}} = \sqrt{\frac{\pi}{\beta A}},</math>
in the formula for {{math|''Z''}}. The mean energy associated with this factor is given by
<math display="block">\langle H_{x} \rangle = - \frac{\partial \log Z_{x}}{\partial \beta} = \frac{1}{2\beta} = \frac{1}{2} k_\text{B} T</math>
as stated by the equipartition theorem.

===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.