Editing Equipartition theorem (section)

===Anharmonic oscillators===
{{See also|Anharmonic oscillator}}

An anharmonic oscillator (in contrast to a simple harmonic oscillator) is one in which the potential energy is not quadratic in the extension {{mvar|q}} (the [[canonical coordinate|generalized position]] which measures the deviation of the system from equilibrium). Such oscillators provide a complementary point of view on the equipartition theorem.<ref name="tolman_1927">{{cite book | last = Tolman | first = RC | author-link = Richard C. Tolman | year = 1927 | title = Statistical Mechanics, with Applications to Physics and Chemistry | url = https://archive.org/details/statisticalmecha00tolm | publisher = Chemical Catalog Company | pages = [https://archive.org/details/statisticalmecha00tolm/page/76 76–77]}}</ref><ref name="terletskii_1971">{{cite book | last = Terletskii | first = YP | year = 1971 | title = Statistical Physics | edition = translated: N. Fröman | publisher = North-Holland | location = Amsterdam | isbn = 0-7204-0221-2 | pages = 83–84 | lccn = 70157006}}</ref> Simple examples are provided by potential energy functions of the form
<math display="block">H_{\mathrm{pot}} = C q^{s},\,</math>
where {{mvar|C}} and {{mvar|s}} are arbitrary [[real number|real constants]]. In these cases, the law of equipartition predicts that
<math display="block">
k_\text{B} T = \left\langle q \frac{\partial H_{\mathrm{pot}}}{\partial q} \right\rangle = 
\langle q \cdot s C q^{s-1} \rangle = \langle s C q^{s} \rangle = s \langle H_{\mathrm{pot}} \rangle.
</math>
Thus, the average potential energy equals {{math|''k''<sub>B</sub>''T''/''s''}}, not {{math|''k''<sub>B</sub>''T''/2}} as for the quadratic harmonic oscillator (where {{math|1=''s'' = 2}}).

More generally, a typical energy function of a one-dimensional system has a [[Taylor expansion]] in the extension {{mvar|q}}:
<math display="block">H_{\mathrm{pot}} = \sum_{n=2}^\infty C_n q^n</math>
for non-negative [[integer]]s {{mvar|n}}. There is no {{math|1=''n'' = 1}} term, because at the [[equilibrium point]], there is no net force and so the first derivative of the energy is zero. The {{math|1=''n'' = 0}} term need not be included, since the energy at the equilibrium position may be set to zero by convention. In this case, the law of equipartition predicts that<ref name="tolman_1927" />
<math display="block">
k_\text{B} T = \left\langle q \frac{\partial H_{\mathrm{pot}}}{\partial q} \right\rangle = 
\sum_{n=2}^{\infty} \langle q \cdot n C_{n} q^{n-1} \rangle = 
\sum_{n=2}^{\infty} n C_{n} \langle q^{n} \rangle.
</math>
In contrast to the other examples cited here, the equipartition formula
<math display="block">
\langle H_{\mathrm{pot}} \rangle = \frac{1}{2} k_\text{B} T - 
\sum_{n=3}^{\infty} \left( \frac{n - 2}{2} \right) C_{n} \langle q^{n} \rangle
</math>
does ''not'' allow the average potential energy to be written in terms of known constants.