Heat capacity and fluctuation-dissipation theorem, meaning of energy fluctuations?

For heat transport in solid, the instantaneous heat flux (resulted from thermal fluctuations) is the sum of atom-resolved energy-density fluctuations times the instantaneous position of the atom,

$textbf{J}(t)=sum_{I=1}^{N} dot{e}_I textbf{R}_{I}$

In the phonon quasi-particle picture, this means the phonon occupations $n_{s}(textbf{q})$ (or $E_s(textbf{q},t)$) of each modes $(s,textbf{q})$ are fluctuating, and

$textbf{J}(t)=sum_{stextbf{q}} E_s(textbf{q},t)textbf{v}_{s}(textbf{q})$

You know $n_{s}(textbf{q})$ and $E_s(textbf{q},t)$ are related with each other by Bose-Einstein distributions