Equipartition theorem

Remarks

Proof idea: in part 2, integrate by parts in $x_k$, using $\partial_k H \, e^{-\beta H} = -\beta^{-1} \partial_k e^{-\beta H}$; the boundary terms vanish and $\partial_k x_m = \delta_{mk}$ remains. Part 1 follows from the one-dimensional Gaussian integral $\int x^2 e^{-\beta a x^2} dx \big/ \int e^{-\beta a x^2} dx = 1/(2 \beta a)$. Example: a classical harmonic oscillator $H = p^2/(2m) + \frac{1}{2} m \omega^2 q^2$ has two quadratic terms, so $\langle H \rangle = k_B T$. A monatomic ideal gas of $N$ atoms has $3N$ quadratic momentum terms, so its mean energy is $\frac{3}{2} N k_B T$. The theorem is classical: for a quantum oscillator with $\hbar \omega \gg k_B T$, the mean energy above the ground state is far below $k_B T$. This explains the low heat capacity of solids at low temperature.

Going deeper

For a student

The general form $\langle x_m \partial_k H \rangle = \delta_{mk} k_B T$ also gives the virial theorem of statistical mechanics when applied to positions. It predicts the heat capacity $\frac{3}{2} N k_B$ of a monatomic gas and the Dulong–Petit value $3 N k_B$ for a solid of $N$ atoms modelled as oscillators. Its failure for radiation and for solids at low temperature was a historical motivation for quantum theory. In the quantum canonical ensemble the theorem holds only in the limit of high temperature.