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I have read that from the fluctuation-dissipation theorem that the heat capacity is proportional to energy fluctuations (or populations fluctuations). In this context what is the meaning of 'energy fluctuations' (since a well defined state has constant energy) and why are they zero at $T=0$?

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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

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

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

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