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

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 $T=0$?

## 1 Answer

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