Editing Equipartition theorem (section)

===Potential energy and harmonic oscillators===
Equipartition applies to [[potential energy|potential energies]] as well as kinetic energies: important examples include [[harmonic oscillator]]s such as a [[spring (device)|spring]], which has a quadratic potential energy
<math display="block">H_{\text{pot}} = \tfrac 1 2 a q^2,\,</math>
where the constant {{mvar|a}} describes the stiffness of the spring and {{mvar|q}} is the deviation from equilibrium. If such a one-dimensional system has mass {{mvar|m}}, then its kinetic energy {{math|''H''<sub>kin</sub>}} is
<math display="block">H_{\text{kin}} = \frac{1}{2}mv^2 = \frac{p^2}{2m},</math>
where {{mvar|v}} and {{math|1=''p'' = ''mv''}} denote the velocity and momentum of the oscillator. Combining these terms yields the total energy<ref name="goldstein_1980" />
<math display="block">H = H_{\text{kin}} + H_{\text{pot}} = \frac{p^2}{2m} + \frac{1}{2} a q^2.</math>
Equipartition therefore implies that in thermal equilibrium, the oscillator has average energy
<math display="block">
\langle H \rangle = 
\langle H_{\text{kin}} \rangle + \langle H_{\text{pot}} \rangle = 
\tfrac{1}{2} k_\text{B} T + \tfrac{1}{2} k_\text{B} T = k_\text{B} T,
</math>
where the angular brackets <math>\left\langle \ldots \right\rangle</math> denote the average of the enclosed quantity,<ref name="huang_1987" />

This result is valid for any type of harmonic oscillator, such as a [[pendulum]], a vibrating molecule or a passive [[electronic oscillator]]. Systems of such oscillators arise in many situations; by equipartition, each such oscillator receives an average total energy {{math|''k''<sub>B</sub>''T''}} and hence contributes {{math|''k''<sub>B</sub>}} to the system's [[heat capacity]]. This can be used to derive the formula for [[Johnson–Nyquist noise]]<ref name="mandl_1971">{{cite book | last = Mandl | first = F | year = 1971 | title = Statistical Physics | publisher = John Wiley and Sons | pages = [https://archive.org/details/statisticalphysi00fman/page/213 213–219] | isbn = 0-471-56658-6 | url = https://archive.org/details/statisticalphysi00fman/page/213 }}</ref> and the [[Dulong–Petit law]] of solid heat capacities. The latter application was particularly significant in the history of equipartition.

[[Image:Vätskefas.png|frame|left|Figure 3. Atoms in a crystal can vibrate about their equilibrium positions in the [[crystal structure|lattice]]. Such vibrations account largely for the [[heat capacity]] of crystalline [[dielectric]]s; with [[metal]]s, [[electron]]s also contribute to the heat capacity.]]