# Proof of equipartition theorem for $i$ not equal to $j$

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

$$\left\langle x_i \frac{∂H}{∂x_j}\right\rangle=δ_{ij}\cdot k_BT.$$

Wikipedia provides proofs of this for a microcanonical ensemble (for any $$i$$, $$j$$) and for a canonical ensemble (ostensibly for $$i=j$$ only). However, the case of a canonical ensemble with $$i$$ not equal to $$j$$ (i.e., $$\langle x_i \, ∂H/∂x_j\rangle =0$$) appears to be missing proof. Is it a simple extension?

Yes, the proof is almost identical to the one on the Wikipedia page, and can be found e.g. in Statistical Mechanics (3rd ed) by RK Pathria and PD Beale, p62. We start with $$\left\langle x_i \frac{\partial H}{\partial x_j} \right\rangle = \frac{\int x_i \frac{\partial H}{\partial x_j} e^{-\beta H} d\Gamma}{\int e^{-\beta H} d\Gamma} .$$ Integration of the numerator by parts with respect to $$x_j$$ gives $$\int x_i \frac{\partial H}{\partial x_j} e^{-\beta H} d\Gamma = \int \left[ \frac{1}{\beta} \int \frac{\partial x_i}{\partial x_j} e^{-\beta H} dx_j \right] d\Gamma'$$ where $$\Gamma'$$ stands for all the coordinates except for $$x_j$$, and a boundary term $$x_i e^{-\beta H}$$ evaluated at $$x_j=\pm\infty$$ has been omitted: it is assumed to vanish on the grounds that $$H$$ becomes infinite whenever $$x_j$$ becomes infinite (or is bounded by the container walls). The important point is that $$\partial x_i/\partial x_j = \delta_{ij}$$ which comes out of the integral. That gives us $$k_BT \, \delta_{ij} \int e^{-\beta H} d\Gamma$$ and the integral cancels with the denominator in the original expression.