This question references C.L.Henley's paper on arxiv. Page 3, section B: Effective free energy and correlations.

There is an ice polarization field $\mathbf{P}(\mathbf{r})$, that has been coarse grained such that it is smoothly varying across the crystal.

The author argues the total free energy (completely entropic in origin) as a function of coarse-grained $\mathbf{P}(\mathbf{r})$:

$$ F_{\mathrm{tot}} ( \mathbf{P}(\mathbf{r})) = \frac{T\kappa}{2v_\mathrm{cell}} \int d^3\mathbf{r} |\mathbf{P}(\mathbf{r})|^2 \tag{2.2} $$ where $\kappa$ is some constant (stiffness), $v_\mathrm{cell}$ is the volume of unit cell, $T$ is temperature.

The Fourier transform of (2.2) gives $$ \tilde{F}_{\mathrm{tot}} = \sum_{\mathbf{k}} \frac{\kappa}{2} |\mathbf{P}(\mathbf{k})|^2 $$

and it might also be relevant to note the probability distribution of the polarization field is a Gaussian distribution (ignoring the ice-rule constraints as in the paper for now): $$ \text{Prob}( \{ \mathbf{P}(\mathbf{r}) \} ) \propto \exp(- F_{\mathrm{tot}} ( \mathbf{P}(\mathbf{r}))/T) \tag{2.4} $$

The part that confuses me is: In the following statement (right after Eq 2.4 in the paper), the author writes:

..., so a naive use of equipartition would give $$ \langle P_\mu (-\mathbf{k}) P_\nu (\mathbf{k}) \rangle = \delta_{\mu\nu}/\kappa $$

but I'm not sure how equipartition comes into play to give you correlations, nor the logic jump that the author seems to make here.