Equipartition theorem (classical canonical ensemble)

Statement

Consider a classical system with phase space phase space and Hamiltonian Hamiltonian $H(x)$, distributed according to the canonical ensemble at inverse temperature $\beta = 1/(k_B T)$, i.e. with density proportional to $e^{-\beta H(x)}$.

If $H$ contains a term that is quadratic in some coordinate $y$ (a position or momentum component), of the form

• $H(x) = \frac{a}{2} y^2 + H_{\mathrm{rest}}(x)$ with $a>0$ and $H_{\mathrm{rest}}$ independent of $y$,

then the canonical expectation satisfies

Equivalently, each independent quadratic term contributes $\frac{1}{2}k_B T$ to the mean energy.

A common general form (under appropriate decay/regularity assumptions) is:

Key hypotheses

• Classical setting with phase space and a well-defined canonical ensemble .
• The coordinate $y$ appears quadratically and the corresponding Gaussian integral is integrable (growth/decay ensures boundary terms vanish).
• Differentiation/integration steps are justified (e.g., dominated convergence).

Conclusion

• For each quadratic degree of freedom, the mean energy contribution is $\frac{1}{2}k_B T$.
• Summing over $m$ independent quadratic terms gives a contribution $\frac{m}{2}k_B T$ to the average internal energy (compare internal energy and temperature ).

Cross-links to definitions

• Canonical ensemble , ensemble average
• canonical partition function
• thermodynamic temperature

Proof idea / significance

Write the canonical expectation as a ratio of integrals with weight $e^{-\beta H}$. For a quadratic coordinate $y$, the $y$-dependence is Gaussian; integrating by parts in $y$ (or evaluating the Gaussian moment explicitly) yields the stated identity. The theorem explains why many classical systems have heat capacities close to “number of quadratic modes times $\frac{1}{2}k_B$,” and also highlights where classical predictions fail (e.g., quantum suppression of modes at low temperature).