# Proof of the generalized equipartition theorem

The generalized equipartition theorem (where variables need not be quadratic) states that if $x_i$ is a canonical variable (position or momentum variable), then

$$\left\langle x_i \frac{\partial \mathcal{H}}{\partial x_j}\right\rangle = \delta_{ij}\ k T$$

where the average $\langle \cdot \rangle$ is taken over an equilibrium probability density $\rho(p,q)$:

$$\langle f(p,q) \rangle = \int dp dq \ \rho(p,q) \ f(p,q)$$

In the most general case this probability density is the canonical ensemble's. For the theorem to hold ergodicity is also required. However, I'm having trouble finding a rigorous prove where the assumptions are explicitly used in the derivation.

Could you provide such a prove, or a reference to a paper/book where it can be found?

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• The proof is essentially identical to the proof of the standard equipartition theorem. What about the proof on the wiki page are you not satisfied by? The assumption of equilibrium and erodicity are simply used to obtain the form of $\rho(p,q)$, which will be covered in book on Stat Mech which you find sufficiently rigorous. – By Symmetry Feb 5 '17 at 16:36
• Have you looked at the Wikipedia article and the links therein? en.m.wikipedia.org/wiki/Equipartition_theorem – Farcher Feb 5 '17 at 16:38
• I have read the proof on the wikipedia page, but I don't see clearly where ergodicity is assumed. I imagine this is related with the Gibbs distribution in the canonical ensemble; however, I'm not an expert in the field, and that's the reason I asked ;) – nabla Feb 5 '17 at 20:32