Editing Equipartition theorem (section)

===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.