Editing Equipartition theorem (section)

===Brownian motion===
[[Image:Wiener process 3d.png|thumb|upright=1.5|Figure 7. Example Brownian motion of a particle in three dimensions.]]
The equipartition theorem can be used to derive the [[Brownian motion]] of a particle from the [[Langevin equation]].<ref name="pathria_1972" /> According to that equation, the motion of a particle of mass {{mvar|m}} with velocity {{math|'''v'''}} is governed by [[Newton's laws of motion|Newton's second law]]
<math display="block">\frac{d\mathbf{v}}{dt} = \frac{1}{m} \mathbf{F} = -\frac{\mathbf{v}}{\tau} + \frac{1}{m} \mathbf{F}_{\mathrm{rnd}},</math>
where {{math|'''F'''<sub>rnd</sub>}} is a random force representing the random collisions of the particle and the surrounding molecules, and where the [[time constant]] τ reflects the [[drag (physics)|drag force]] that opposes the particle's motion through the solution. The drag force is often written {{math|1='''F'''<sub>drag</sub> = −''γ'''''v'''}}; therefore, the time constant {{math|''τ''}} equals {{math|''m''/''γ''}}.

The dot product of this equation with the position vector {{math|'''r'''}}, after averaging, yields the equation
<math display="block">
\left\langle \mathbf{r} \cdot \frac{d\mathbf{v}}{dt} \right\rangle + 
\frac{1}{\tau} \langle \mathbf{r} \cdot \mathbf{v} \rangle = 0
</math>
for Brownian motion (since the random force {{math|'''F'''<sub>rnd</sub>}} is uncorrelated with the position {{math|'''r'''}}). Using the mathematical identities
<math display="block">
\frac{d}{dt} \left( \mathbf{r} \cdot \mathbf{r} \right) = 
\frac{d}{dt} \left( r^{2} \right) = 2 \left( \mathbf{r} \cdot \mathbf{v} \right)
</math>
and
<math display="block">\frac{d}{dt} \left( \mathbf{r} \cdot \mathbf{v} \right) = v^{2} + \mathbf{r} \cdot \frac{d\mathbf{v}}{dt},</math>
the basic equation for Brownian motion can be transformed into
<math display="block">
\frac{d^{2}}{dt^{2}} \langle r^{2} \rangle + \frac{1}{\tau} \frac{d}{dt} \langle r^{2} \rangle = 
2 \langle v^{2} \rangle = \frac{6}{m} k_\text{B} T,
</math>
where the last equality follows from the equipartition theorem for translational kinetic energy:
<math display="block">
\langle H_{\mathrm{kin}} \rangle = \left\langle \frac{p^2}{2m} \right\rangle = \langle \tfrac{1}{2} m v^{2} \rangle = \tfrac{3}{2} k_\text{B} T.
</math>
The above [[differential equation]] for <math>\langle r^2\rangle</math> (with suitable initial conditions) may be solved exactly:
<math display="block">\langle r^{2} \rangle = \frac{6k_\text{B} T \tau^{2}}{m} \left( e^{-t/\tau} - 1 + \frac{t}{\tau} \right).</math>
On small time scales, with {{math|''t'' ≪ ''τ''}}, the particle acts as a freely moving particle: by the [[Taylor series]] of the [[exponential function]], the squared distance grows approximately ''quadratically'':
<math display="block">\langle r^{2} \rangle \approx \frac{3k_\text{B} T}{m} t^2 = \langle v^{2} \rangle t^{2}.</math>
However, on long time scales, with {{math|''t'' ≫ ''τ''}}, the exponential and constant terms are negligible, and the squared distance grows only ''linearly'':
<math display="block">\langle r^{2} \rangle \approx \frac{6k_\text{B} T\tau}{m} t = \frac{6 k_\text{B} T t}{\gamma}.</math>
This describes the [[diffusion]] of the particle over time. An analogous equation for the rotational diffusion of a rigid molecule can be derived in a similar way.